Shen Qu, 918881147
STAT564 Fall 2018 Final Exam
12/05/2018
Question 1 (Show main steps of your work to get full points)
The \((X'X)^{-1}\) for the \(y=β_0+β_1 x_1+β_2 x_2+β_3 x_3+β_4 x_4+β_5 x_5+β_6 x_6+ε\) is given below.
If MSE = 1.395 and n = 38 , compute the (Keep 4 or more decimal places, DO NOT round in the intermediate steps)
se(β ̂_4)
\[se(\mathbf{\hat\beta_4})=\sqrt{MSE\times C_{55}}=\sqrt{1.395\times0.069}=0.3102499\]
- Cov(β ̂_2,β ̂_4)
\[Cov(\mathbf{\hat\beta_2,\hat\beta_4})=MSE\times C_{35}=1.395\times(-0.035)=-0.048825\]
- Cor(β ̂_2,β ̂_4 )
\[se(\mathbf{\hat\beta_2})=\sqrt{MSE\times C_{33}}=\sqrt{1.395\times0.067}=0.3057205\]
\[Cor(\mathbf{\hat\beta_2,\hat\beta_4})=\frac{Cov(\mathbf{\hat\beta_2,\hat\beta_4})}{se(\mathbf{\hat\beta_2})se(\mathbf{\hat\beta_4})}=\frac{-0.048825}{0.3057205\times0.3102499}=-0.5147615\]
- Without computing anything, explain which estimator is the most consistent.
\(C_{66}=0.058\) has the smallest value. \(\hatβ_5\) has the the least variance and the most consistent among the estimators.
- Without computing anything , list the pair(s) of estimators that are positively correlated. Provide a reason.
According to the \((X'X)^{(-1)}\),
\(C_{13},\ C_{17},\ C_{24},\ C_{25},\ C_{67}\) are positive.
The positively correlated pairs of parameters are
\(\hatβ_0\) and \(\hatβ_2\), \(\hatβ_0\) and \(\hatβ_6\), \(\hatβ_1\) and \(\hatβ_3\), \(\hatβ_1\) and \(\hatβ_4\), \(\hatβ_5\) and \(\hatβ_6\).
Consider the following hypothesis: \(H_0: β_1=2β_3,β_2=β_3,β_5=0\)
Report the T matrix, β vector and c vector along with their dimensions, and the rank of T matrix for testing the above hypothesis.
\[ \mathbf{T}=\begin{bmatrix} 0 & 1 & 0 & -2 & 0 & 0& 0 \\ 0 & 0 & 1 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \end{bmatrix}_{3\times7} \mathbf{β}=\begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \beta_3 \\ \beta_4 \\ \beta_5 \\ \beta_6 \end{bmatrix}_{7\times1} \mathbf{C}=\begin{bmatrix} 0 \\ 0 \\ 0\end{bmatrix}_{3\times1} rank(T)=3 \]
- Report the values of numerator and denominator degrees of freedom for the corresponding F test. For F test, the numerator is MSR while denominator is MSE, thus
In this hypothesis,\(y=β_0+2β_3x_1+β_3x_2+β_3x_3+β_4x_4+0x_5+β_6x_6+ε=β_0+β_3(2x_1+x_2+x_3)+β_4x_4+β_6x_6+ε\)
The value of numerator is \(r=df_{Reduced}-df_{Full}=n-(3+1)-[n-(6+1)]=3\)
The denominator degrees of freedom is \(df_{Full}=n-(k+1)=38-(6+1)=31\)
- Show the following equation is an alternative form of the sum of squares of regreesion or model (SSR).
\[SSR=\sum_{i=1}^n(\hat y_i-\bar y)^2=\sum_{i=1}^n(\hat y_i^2-2\hat y_i\bar y+\bar y^2)=\sum_{i=1}^n\hat y_i^2-2\bar y\sum_{i=1}^n\hat y_i+\sum_{i=1}^n\bar y^2\]
\[=\sum_{i=1}^n\hat y_i^2-2\bar yn\frac{\sum_{i=1}^n\hat y_i}n+n\bar y^2=\sum_{i=1}^n\hat y_i^2-2\bar yn\bar y+n\bar y^2=\sum_{i=1}^n\hat y_i^2-n\bar y^2\]
Question 2 (Use software to analyze the given data)
The data in the WaterFlow file are simulated data on peak rate of flow (in cfs) of water from six watersheds following storm episodes. The predictors are:
x1 : Area of watershed (mi2) x2 : Area impervious to water (mi2)
x3 : Average slope of watershed (percent)
x4 : Longest stream flow in watershed (1000s of feet)
x5 : surface absorbency index, (0= complete absorbency, 100=no absorbency)
x6 : estimated soil storage capacity (inches of water)
x7 : Infiltration rate of water into soil (inches/hour)
x8 : Rainfall (inches)
x9 : Time period during which rainfall exceeded ¼ inch/hour
- Create the matrix of scatterplots and compute the correlation matrix for all the variables. Copy and paste them here.
- Based on scatterplots and correlation, explain which predictors are significantly related to (most likely to contribute to the variation in) the response variable.
Based on scatterplots and correlation, X2(0.666),X7(0.668),X1(0.781),X4(0.866) have medium to strong positive linear relationship to the response variable (Correlation coefficient is more than 0.6). X5(-0.62) have medium negative linear relationship to the response variable.
- Fit the full model.
\[\hat y=292.561-203.144X_1+ 1055.782X_2-49.24X_3+209.762X_4-10.197X_5-24.558X_6+142.778X_7+511.713X_8-301.872X_9\]
- Explain whether the overall model is significant at 5% significance level.
The fitted model is statistically significant at 5% significance level (\(p-value=9.744\times^{-06}\)). But most of the coefficients are not significent. This model is not the best fitted model.
- Explain whether assumptions of random errors and model are satisfied. If there is a violation of those, then suggest reasonable methods to correct them.
- Residual Diagnostics: Use plots to examine residuals to validate OLS assumptions
There is some violation of assumptions about the errors:
On the residual plot, there is a funnel pattern.
On the outlier and leverage plot, there are two outliers.
On the qq plot, most of points follow approximately straight line but have some positive skew.
- Suggestion: Transform and other diagnostics.
I suggest using natural log of response to make a variance-stabilizing transformations.
Other diagnostics of heteroskedasticity, variable selection, measures of influence also should be considered.
- How much of the sum of squares is explained by rainfall, given that all the other regression coefficients are in the model?
Accroding to the F test, the partial sum of squares explained by rainfall is 2209825, given that all the other regression coefficients are in the model.
- Explain whether there is a problem of multicollinearity.
- Collinearity diagnostics:
The model does have serious problems of multicollinearity. The VIF of variables X4 (105.754708), X1 (101.859709), X3 (31.446394), X7(20.53505) are larger than 10.
- Interpret the estimated coefficient of rainfall predictor of the full model using question context.
Coefficient of 511.713 suggests the peak rate of flow increases by 511.713 cubic feet per second when the rainfall increases by 1 inch and other variables are constants.
- Create a new variable using natural log of response. Then fit the full model using this new variable as response.
\[log(\hat y)=3.402256-0.013532X_1-1.023664X_2+0.177966X_3+0.108788X_4\] \[-0.009622X_5-0.389474X_6+4.233475X_7+0.63007X_8-0.462276X_9\]
- Explain whether the overall model is significant at 5% significance level.
The fitted model is statistically significant at 5% significance level (\(p-value=7.513\times10^{-11}\)). But most of the coefficients are not significent. This model is not the best fitted model.
- Explain whether there is a problem of multicollinearity.
The model still has serious problems of multicollinearity. The variance-stabilizing transformations does not change the value of VIF.
It will be important to solve multicollinearity. However, X7 (20.53505), X1 (101.859709), and X4 (105.754708) have medium to strong positive linear relationship to the response variable. It is also dangerous to remove these variables. We should have more diagnostics and comparisons.
- If you wanted to simplify this full model, explain which predictor you would eliminate first.
- VIF
If just considering the VIF, X4 (105.754708) or X1 (101.859709) with largest VIF values is the first to remove.
- Correlation with y
However, according to the correlation coefficients, both X4and X1 is strongly correlated with y (\(Cor_{y,x_4}=0.866,Cor_{y,x_1}=0.781\)) .
The textbook suggest that the general approaches for dealing with multicollinearity include collecting additional data, model respecification (redefine the regressors, variable elimination), estimation methods (Ridge Regression, Principal-Component Regression). “Variable elimination is often a highly effective technique. However, it may not provide a satisfactory solution if the regressors dropped from the model have significant explanatory power relative to the response y. That is, eliminating regressors to reduce multicollinearity may damage the predictive power of the model.” (Montgomery et al., 2012. p.304) In this way, the third multicollinear X3(31.446394) with a weak relationship with y (0.205) should be removed, or even X6(0.0453).
- Correlation with each other
According to the variable names of X4, X1, and X3, they are geographic variables. Predictor X1 is the area of watershed while X4 is the longest stream flow in watershed, x3 is the average slope of watershed. For the given 6 watersheds, X1 and X4 are strongly related. A high correlation (0.921) is expected between these two variables. But X3 is not significently related with X1(-0.078) or X4(0.245). Removing X3 might lose some irreplacable infromation. I
Actrually, I don’t agree remove any predictor in this stage. Removing any predictor can draw down the VIF significently. After elimination regression, the multicollinearity dissapeared in all the models. We should gather sufficient evidents before removing any predictor.
- Use the forward selection method to find the best model (use α=0.15) and report the final fitted model with estimated coefficients here.
Use Stepwise Forward Regression based on p values (use α=0.15)
\[\hat y=2.872+0.168X_3+0.122X_4+3.106X_7\]
Use Stepwise AIC Forwardd Regression
\[\hat y=2.692+0.184X_3+0.109X_4-0.368X_6+4.085X_7+0.612X_8-0.448X_9\]
- Use the backward elimination method to find the best model (use α=0.05) and report the final fitted model with estimated coefficients here.
Stepwise Backward Regression based on p values (use α=0.05) and Stepwise AIC Backward Regression have same results.
\[\hat y=2.692+0.184X_3+0.109X_4-0.368X_6+4.085X_7+0.612X_8-0.448X_9\]
- Use best subsets method (6 models from each size) to find the best model for these data and report the final fitted model with estimated coefficients here.
Best subsets method gives a same model.
\[\hat y=2.692+0.184X_3+0.109X_4-0.368X_6+4.085X_7+0.612X_8-0.448X_9\]
- If the final models in the previous 3 methods are different, compare their model adequacy and suggest one best model.
| Method | By | Keep | Remove |
|---|---|---|---|
| Stepwise Forward | P=0.15 | X3, X4, X7 | X1,X2,X5,X6,X8,X9 |
| Stepwise Forward | AIC | X3,X4,X6,X7,X8,X9 | X1,X2,X5 |
| Stepwise Backward | P=0.05 | X3,X4,X6,X7,X8,X9 | X1,X2,X5 |
| Stepwise Backward | AIC | X3,X4,X6,X7,X8,X9 | X1,X2,X5 |
| Stepwise Both | P | X3, X4, X7 | X1,X2,X5,X6,X8,X9 |
| Stepwise Both | AIC | X3,X4,X6,X7,X8,X9 | X1,X2,X5 |
| Best Subset | / | X3,X4,X6,X7,X8,X9 | X1,X2,X5 |
| all possible | / | X3,X4,X6,X7,X8,X9 | X1,X2,X5 |
Both models solved the problem of multicollinearity (VIF <10), and small P-values for F test. They don’t have serious violation of assumptions about the errors (There is no significant pattern on the plot of studentized residuals versus predicted values from the model with only one predictor. The partial regression plots do not show nonlinear patterns. The points follow approximately straight line on the qq plot). Both of Correlation between observed residuals and expected residuals under normality.The 6-predictor model got 0.9837263 P-value while the 6-predictor model got 0.9856766.
| Model | VIF | F | P-value(F) | MSR | MSE | \(R_{adjusted}^2\) | \(R_{Predict}^2\) | P-value(t) | Residuals Plots |
|---|---|---|---|---|---|---|---|---|---|
| 3-4-7 | <10 | 70.378 | 0.0000 | 21.188 | 0.301 | 0.878 | 0.854 | Max=0.054 | Good enough |
| 3-4-6-7-8-9 | <10 | 68.16 | 0.0000 | 11.265 | 0.165 | 0.933 | 0.908 | Max=0.019 | Good enough |
However, comparing to the 3-variable model, the 6-variable model has a higher (about by 6%) adjusted R square and higher (about by 5%) prediction R-square, which means it shows stronger predictive capability. All the coeficients in 6-predictors model are statistically significant higher than 98% significance level (the maximum p-values are 0.019, respectively). In the 3-variable model, X7 get a high p-value (0.054) which means not significant at 5% significance level. If we change the p-value as the parameter of forward selection, the same model will happened between \(\alpha\) equal 0.6 and 0.17. Further, considering the context, X8 and X9 are variables of precipitation. The 3-predictor model mean the peak flow is irrelevant with precipitation. It doesn’t make sense. Therefore, the best model will be the model with 6 predictors.
- Provide complete ANOVA table for the best model. Provide partial sum of squares, estimated coefficients, standard errors, p-values, 95% Bonferroni joint confidence intervals for the coefficients of the best model. Provide in a tabular form clearly.
| Model Summary | |||
|---|---|---|---|
| R | 0.973 | RMSE (Root Mean Square Error) | 0.407 |
| R-Squared | 0.947 | Coef. Var | 6.385 |
| Adj. R-Squared | 0.933 | MSE (Mean Square Error) | 0.165 |
| Pred R-Squared | 0.908 | MAE (Mean Absolute Error) | 0.273 |
| ANOVA | |||||
|---|---|---|---|---|---|
| Sum of Squares | DF | Mean Square | F | p-value | |
| Regression | 67.591 | 6 | 11.265 | 68.16 | \(1.717\times10^{-13}\) |
| Residual | 3.801 | 23 | 0.165 | ||
| Total | 71.393 | 29 |
| Parameter | Estimates | ||||||
|---|---|---|---|---|---|---|---|
| model | Estimated coefficients | Partial SS | Std. Error | t test | p-value | 2.5 % | 97.5 % |
| (Intercept) | 2.69180 | / | 0.445 | 6.046 | \(3.63\times10{-06}\) | 1.77080533 | 3.61278901 |
| X3 | 0.18384 | 5.37 | 0.032 | 5.698 | \(8.41\times10{-06}\) | 0.11709482 | 0.25059326 |
| X4 | 0.10905 | 2.98 | 0.026 | 4.244 | 0.000306 | 0.05589763 | 0.16220302 |
| X6 | -0.36752 | 1.05 | 0.146 | -2.526 | 0.018898 | -0.66855123 | -0.06648201 |
| X7 | 4.08497 | 1.87 | 1.213 | 3.367 | 0.002662 | 1.57533753 | 6.59460252 |
| X8 | 0.61161 | 3.52 | 0.133 | 4.614 | 0.000122 | 0.33738118 | 0.88582991 |
| X9 | -0.44764 | 2.83 | 0.108 | -4.135 | 0.000402 | -0.67159465 | -0.22368135 |
- How much variation in the response is explained by the best model after taking number of data and regression coefficients in to account?
By SSR equal 67.591 and SSE equal 3.801, the adjusted R-squared is 0.9329. About 93.29% variation in the response is explained by the best model.
- Report the PRESS statistic of the best model.
The value of PRESS is 6.538275. This model explains 90.8% of variation in predicting the peak rate of flow (in cfs) of water from six watersheds following storm episodes.
- Report the complete code along with output here.
Singh (1972) used linear models with a logarithm transformation of the variables. We retained the following where the dependent variables can either be total storm flow volume (Qt) in mm, quick flow volume (Qf) in mm or peak flow (Qpk) in m3 sec−1 km−2. Independent variables were storm rainfall (P) in mm, initial flow (Qi) in mm h−1, rainfall frequency (Fp), the inverse of rainfall duration, in h−1 and a dummy variable (CC) representing the treatment effect on basin 7A. CC was 0 and 1 for the calibration (1967–1992) and treated (1994–1998) periods, respectively. β0 to β4 are regression coefficients of the independent variables. All interactions between the independent variables were also tested for significance at α=0.10.
\[ln(Dependent\ Variable)=β_0+β_1lnP+β_2lnQ+β_3lnFP+β_4CC+Interactions+\varepsilon\]
The significance of the regression coefficients (being different from 0) in the models has been tested with a t-test procedure at α=0.10 using the GLM procedure of the SAS system for Windows (SAS Institute, Inc., 1989). A regression coefficient significantly different from 0 for the variable CC indicates that the treatment had a significant effect on the dependent variable. Normality of residuals has been tested using the Shapiro–Wilk test. Selection criteria for all events were the same as for the paired basins approach. However, the rainfall events following night-frost, which may caused localized surface runoff on ice as observed by Prevost et al. (1990), were omitted since these events are too rare to be well represented during the calibration and post-treatment periods. Hence, all events within three weeks following the end of the snowmelt period were not retained. The end of the snowmelt period was obtained from observations in a standard forest snow line at Montmorency Forest. (Guillemette et al., 2005)
- Residual Diagnostics:
Includes plots to examine residuals to validate OLS assumptions
There is no violation of assumptions about the errors (no pattern on residual plots and points follow approximately straight line on the qq plot).
- Variable selection:
Differnt variable selection procedures such as all possible regression, best subset regression, stepwise regression, stepwise forward regression and stepwise backward regression
- Heteroskedasticity:
Tests for heteroskedasticity include bartlett test, breusch pagan test, score test and f test
- Measures of influence:
Use different plots to detect and identify influential observations
- Collinearity diagnostics:
VIF, Tolerance and condition indices to detect collinearity and plots for assessing mode fit and contributions of variables
x1 : Area of watershed (mi2)
x8 : Rainfall (inches)
x9 : Time period during which rainfall exceeded ¼ inch/hour
x4 : Longest stream flow in watershed (1000s of feet)
x3 : Average slope of watershed (percent)
x2 : Area impervious to water (mi2)
x5 : surface absorbency index, (0= complete absorbency, 100=no absorbency)
x7 : Infiltration rate of water into soil (inches/hour)
x6 : estimated soil storage capacity (inches of water)
Full model
eliminated model
- Residual Diagnostics:
Includes plots to examine residuals to validate OLS assumptions
There is no violation of assumptions about the errors (no pattern on residual plots and points follow approximately straight line on the qq plot).
Residual QQ Plot Residual Normality Test Residual vs Fitted Values Plot Residual Histogram
- Variable selection:
Differnt variable selection procedures such as all possible regression, best subset regression, stepwise regression, stepwise forward regression and stepwise backward regression
- Heteroskedasticity:
Tests for heteroskedasticity include bartlett test, breusch pagan test, score test and f test
Bartlett Test Breusch Pagan Test Score Test F Test
- Measures of influence:
Use different plots to detect and identify influential observations
Cook’s D Bar Plot Cook’s D Chart DFBETAs Panel DFFITs Plot Studentized Residual Plot Standardized Residual Chart Studentized Residuals vs Leverage Plot Deleted Studentized Residual vs Fitted Values Plot Hadi Plot Potential Residual Plot
[1]: Montgomery, D. C., Peck, E. A., & Vining, G. G. (2012). Introduction to linear regression analysis (Vol. 821). John Wiley & Sons.
[2]: Guillemette, F., Plamondon, A. P., Prévost, M., & Lévesque, D. (2005). Rainfall generated stormflow response to clearcutting a boreal forest: peak flow comparison with 50 world-wide basin studies. Journal of hydrology, 302(1-4), 137-153.
(a) The matrix of scatterplots and the correlation matrix
library(tidyverse)
table_wf <- read_table2("WaterFlow.txt")
library(GGally)
ggpairs(data=table_wf[c(1:10)])(c) The fitted full model
# build the model
model_wf_full <- lm(y ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9, data=table_wf)
model_wf_full%>% summary()##
## Call:
## lm(formula = y ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9,
## data = table_wf)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1404.21 -318.77 74.73 266.66 1274.30
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 292.56 4428.62 0.066 0.9480
## X1 -203.14 410.27 -0.495 0.6259
## X2 1055.78 9833.70 0.107 0.9156
## X3 -49.24 156.20 -0.315 0.7558
## X4 209.76 162.05 1.294 0.2103
## X5 -10.20 51.09 -0.200 0.8438
## X6 -24.56 303.53 -0.081 0.9363
## X7 142.78 3288.44 0.043 0.9658
## X8 511.71 209.74 2.440 0.0241 *
## X9 -301.87 172.00 -1.755 0.0945 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 609.3 on 20 degrees of freedom
## Multiple R-squared: 0.8214, Adjusted R-squared: 0.741
## F-statistic: 10.22 on 9 and 20 DF, p-value: 9.744e-06Anova(model_wf_full)## Anova Table (Type II tests)
##
## Response: y
## Sum Sq Df F value Pr(>F)
## X1 91022 1 0.2452 0.62589
## X2 4279 1 0.0115 0.91557
## X3 36893 1 0.0994 0.75585
## X4 622091 1 1.6756 0.21025
## X5 14790 1 0.0398 0.84381
## X6 2430 1 0.0065 0.93632
## X7 700 1 0.0019 0.96580
## X8 2209825 1 5.9523 0.02414 *
## X9 1143622 1 3.0804 0.09455 .
## Residuals 7425127 20
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1(c) ii Residual diagnostics
#Model Fit Assessment
ols_plot_diagnostics(model_wf_full)
# Part & Partial Correlations
ols_test_correlation(model_wf_full) # Correlation between observed residuals and expected residuals under normality.## [1] 0.9710713# Residual Normality Test
ols_test_normality(model_wf_full) # Test for detecting violation of normality assumption. #If p-value is bigger, then no problem of non-normality ### -----------------------------------------------
## Test Statistic pvalue
## -----------------------------------------------
## Shapiro-Wilk 0.9589 0.2898
## Kolmogorov-Smirnov 0.1423 0.5314
## Cramer-von Mises 2.5333 0.0000
## Anderson-Darling 0.5169 0.1748
## -----------------------------------------------(c) iii The partial regression and nonlinear diagnostics
#Lack of Fit F Test
ols_pure_error_anova(lm(y~X8, data = table_wf))## Lack of Fit F Test
## ---------------
## Response : y
## Predictor: X8
##
## Analysis of Variance Table
## -------------------------------------------------------------------------
## DF Sum Sq Mean Sq F Value Pr(>F)
## -------------------------------------------------------------------------
## X8 1 4616882.92 4616882.92 5.795558 0.02290414
## Residual 28 36951252.44 1319687.59
## Lack of fit 21 31374881.28 1494041.97 1.875466 0.2003839
## Pure Error 7 5576371.17 796624.45
## -------------------------------------------------------------------------# Variable Contributions
ols_plot_added_variable(model_wf_full)
# Residual Plus Component Plot
ols_plot_comp_plus_resid(model_wf_full)
(c) iv Collinearity diagnostics
# for full model
ols_vif_tol(model_wf_full)## # A tibble: 9 x 3
## Variables Tolerance VIF
## <chr> <dbl> <dbl>
## 1 X1 0.00982 102.
## 2 X2 0.133 7.52
## 3 X3 0.0318 31.4
## 4 X4 0.00946 106.
## 5 X5 0.103 9.68
## 6 X6 0.433 2.31
## 7 X7 0.0487 20.5
## 8 X8 0.182 5.50
## 9 X9 0.174 5.75(d) The fitted log model
# build full log model
model_wf_full_log <- lm(log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9, data=table_wf)
summary(model_wf_full_log)##
## Call:
## lm(formula = log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 +
## X9, data = table_wf)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.95298 -0.20764 0.01499 0.18100 0.67539
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.402256 3.150312 1.080 0.293006
## X1 -0.013532 0.291845 -0.046 0.963477
## X2 -1.023664 6.995235 -0.146 0.885120
## X3 0.177966 0.111113 1.602 0.124908
## X4 0.108788 0.115272 0.944 0.356560
## X5 -0.009622 0.036341 -0.265 0.793898
## X6 -0.389474 0.215916 -1.804 0.086345 .
## X7 4.233475 2.339245 1.810 0.085387 .
## X8 0.630070 0.149200 4.223 0.000418 ***
## X9 -0.462276 0.122350 -3.778 0.001181 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4334 on 20 degrees of freedom
## Multiple R-squared: 0.9474, Adjusted R-squared: 0.9237
## F-statistic: 40 on 9 and 20 DF, p-value: 7.513e-11Anova(model_wf_full)## Anova Table (Type II tests)
##
## Response: y
## Sum Sq Df F value Pr(>F)
## X1 91022 1 0.2452 0.62589
## X2 4279 1 0.0115 0.91557
## X3 36893 1 0.0994 0.75585
## X4 622091 1 1.6756 0.21025
## X5 14790 1 0.0398 0.84381
## X6 2430 1 0.0065 0.93632
## X7 700 1 0.0019 0.96580
## X8 2209825 1 5.9523 0.02414 *
## X9 1143622 1 3.0804 0.09455 .
## Residuals 7425127 20
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#Model Fit Assessment
# ols_plot_diagnostics(model_wf_full_log)
# Part & Partial Correlations
# ols_test_correlation(model_wf_full_log) # Correlation between observed residuals and expected residuals under normality.
# Residual Normality Test
# ols_test_normality(model_wf_full_log) # Test for detecting violation of normality assumption. #If p-value is bigger, then no problem of non-normality #
library(dplyr)(d) (2) Collinearity diagnostics
## Start: AIC=-42.32
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9
##
## Df Sum of Sq RSS AIC
## - X1 1 0.0004 3.7577 -44.322
## - X2 1 0.0040 3.7613 -44.293
## - X5 1 0.0132 3.7705 -44.220
## - X4 1 0.1673 3.9246 -43.018
## <none> 3.7573 -42.325
## - X3 1 0.4819 4.2392 -40.705
## - X6 1 0.6113 4.3686 -39.803
## - X7 1 0.6153 4.3726 -39.775
## - X9 1 2.6819 6.4392 -28.164
## - X8 1 3.3503 7.1076 -25.201
##
## Step: AIC=-44.32
## log(y) ~ X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9
##
## Df Sum of Sq RSS AIC
## - X2 1 0.0110 3.7686 -46.234
## - X5 1 0.0267 3.7844 -46.110
## <none> 3.7577 -44.322
## - X6 1 1.0447 4.8023 -38.963
## - X7 1 1.5520 5.3097 -35.950
## - X4 1 1.8469 5.6046 -34.328
## - X9 1 2.8341 6.5918 -29.461
## - X8 1 3.4848 7.2425 -26.637
## - X3 1 5.0955 8.8532 -20.613
##
## Step: AIC=-46.23
## log(y) ~ X3 + X4 + X5 + X6 + X7 + X8 + X9
##
## Df Sum of Sq RSS AIC
## - X5 1 0.0327 3.8013 -47.975
## <none> 3.7686 -46.234
## - X6 1 1.0375 4.8061 -40.939
## - X4 1 1.8741 5.6428 -36.125
## - X7 1 1.9036 5.6722 -35.968
## - X9 1 2.8353 6.6040 -31.406
## - X8 1 3.4744 7.2430 -28.635
## - X3 1 5.1264 8.8951 -22.471
##
## Step: AIC=-47.98
## log(y) ~ X3 + X4 + X6 + X7 + X8 + X9
##
## Df Sum of Sq RSS AIC
## <none> 3.8013 -47.975
## - X6 1 1.0542 4.8555 -42.632
## - X7 1 1.8739 5.6752 -37.953
## - X9 1 2.8256 6.6270 -33.302
## - X4 1 2.9771 6.7784 -32.624
## - X8 1 3.5182 7.3195 -30.320
## - X3 1 5.3653 9.1666 -23.569## Start: AIC=-44.32
## log(y) ~ X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9
##
## Df Sum of Sq RSS AIC
## - X2 1 0.0110 3.7686 -46.234
## - X5 1 0.0267 3.7844 -46.110
## <none> 3.7577 -44.322
## - X6 1 1.0447 4.8023 -38.963
## - X7 1 1.5520 5.3097 -35.950
## - X4 1 1.8469 5.6046 -34.328
## - X9 1 2.8341 6.5918 -29.461
## - X8 1 3.4848 7.2425 -26.637
## - X3 1 5.0955 8.8532 -20.613
##
## Step: AIC=-46.23
## log(y) ~ X3 + X4 + X5 + X6 + X7 + X8 + X9
##
## Df Sum of Sq RSS AIC
## - X5 1 0.0327 3.8013 -47.975
## <none> 3.7686 -46.234
## - X6 1 1.0375 4.8061 -40.939
## - X4 1 1.8741 5.6428 -36.125
## - X7 1 1.9036 5.6722 -35.968
## - X9 1 2.8353 6.6040 -31.406
## - X8 1 3.4744 7.2430 -28.635
## - X3 1 5.1264 8.8951 -22.471
##
## Step: AIC=-47.98
## log(y) ~ X3 + X4 + X6 + X7 + X8 + X9
##
## Df Sum of Sq RSS AIC
## <none> 3.8013 -47.975
## - X6 1 1.0542 4.8555 -42.632
## - X7 1 1.8739 5.6752 -37.953
## - X9 1 2.8256 6.6270 -33.302
## - X4 1 2.9771 6.7784 -32.624
## - X8 1 3.5182 7.3195 -30.320
## - X3 1 5.3653 9.1666 -23.569## Start: AIC=-44.29
## log(y) ~ X1 + X3 + X4 + X5 + X6 + X7 + X8 + X9
##
## Df Sum of Sq RSS AIC
## - X1 1 0.0073 3.7686 -46.234
## - X5 1 0.0370 3.7983 -45.999
## <none> 3.7613 -44.293
## - X4 1 0.3141 4.0754 -43.887
## - X6 1 0.7115 4.4729 -41.095
## - X3 1 0.7775 4.5388 -40.656
## - X7 1 1.2667 5.0280 -37.585
## - X9 1 2.7122 6.4735 -30.005
## - X8 1 3.4001 7.1614 -26.975
##
## Step: AIC=-46.23
## log(y) ~ X3 + X4 + X5 + X6 + X7 + X8 + X9
##
## Df Sum of Sq RSS AIC
## - X5 1 0.0327 3.8013 -47.975
## <none> 3.7686 -46.234
## - X6 1 1.0375 4.8061 -40.939
## - X4 1 1.8741 5.6428 -36.125
## - X7 1 1.9036 5.6722 -35.968
## - X9 1 2.8353 6.6040 -31.406
## - X8 1 3.4744 7.2430 -28.635
## - X3 1 5.1264 8.8951 -22.471
##
## Step: AIC=-47.98
## log(y) ~ X3 + X4 + X6 + X7 + X8 + X9
##
## Df Sum of Sq RSS AIC
## <none> 3.8013 -47.975
## - X6 1 1.0542 4.8555 -42.632
## - X7 1 1.8739 5.6752 -37.953
## - X9 1 2.8256 6.6270 -33.302
## - X4 1 2.9771 6.7784 -32.624
## - X8 1 3.5182 7.3195 -30.320
## - X3 1 5.3653 9.1666 -23.569## Start: AIC=-40.7
## log(y) ~ X1 + X2 + X4 + X5 + X6 + X7 + X8 + X9
##
## Df Sum of Sq RSS AIC
## - X7 1 0.1337 4.3729 -41.773
## - X6 1 0.1858 4.4250 -41.418
## <none> 4.2392 -40.705
## - X2 1 0.2995 4.5388 -40.656
## - X5 1 1.1498 5.3891 -35.505
## - X9 1 3.5480 7.7872 -24.461
## - X8 1 4.0900 8.3292 -22.443
## - X1 1 4.6140 8.8532 -20.613
## - X4 1 13.7481 17.9873 0.654
##
## Step: AIC=-41.77
## log(y) ~ X1 + X2 + X4 + X5 + X6 + X8 + X9
##
## Df Sum of Sq RSS AIC
## - X6 1 0.1369 4.5098 -42.849
## <none> 4.3729 -41.773
## - X2 1 0.6577 5.0306 -39.570
## - X5 1 1.0236 5.3965 -37.464
## - X9 1 3.4161 7.7890 -26.455
## - X8 1 3.9564 8.3293 -24.442
## - X1 1 4.7933 9.1662 -21.570
## - X4 1 13.8200 18.1929 -1.005
##
## Step: AIC=-42.85
## log(y) ~ X1 + X2 + X4 + X5 + X8 + X9
##
## Df Sum of Sq RSS AIC
## <none> 4.5098 -42.849
## - X2 1 0.6110 5.1208 -41.037
## - X5 1 0.8871 5.3969 -39.461
## - X9 1 3.2799 7.7896 -28.452
## - X8 1 3.8347 8.3444 -26.388
## - X1 1 5.0057 9.5155 -22.448
## - X4 1 15.9600 20.4698 0.533## Start: AIC=-43.02
## log(y) ~ X1 + X2 + X3 + X5 + X6 + X7 + X8 + X9
##
## Df Sum of Sq RSS AIC
## - X5 1 0.0457 3.9703 -44.671
## - X2 1 0.1508 4.0754 -43.887
## <none> 3.9246 -43.018
## - X1 1 1.6800 5.6046 -34.328
## - X6 1 2.1914 6.1160 -31.708
## - X9 1 2.5158 6.4404 -30.158
## - X8 1 3.1937 7.1183 -27.156
## - X7 1 4.3217 8.2463 -22.743
## - X3 1 14.0627 17.9873 0.654
##
## Step: AIC=-44.67
## log(y) ~ X1 + X2 + X3 + X6 + X7 + X8 + X9
##
## Df Sum of Sq RSS AIC
## - X2 1 0.1126 4.0829 -45.832
## <none> 3.9703 -44.671
## - X6 1 2.5195 6.4898 -31.929
## - X9 1 2.7581 6.7284 -30.846
## - X1 1 2.7838 6.7541 -30.731
## - X8 1 3.6308 7.6011 -27.187
## - X7 1 4.2769 8.2472 -24.740
## - X3 1 24.3256 28.2959 12.246
##
## Step: AIC=-45.83
## log(y) ~ X1 + X3 + X6 + X7 + X8 + X9
##
## Df Sum of Sq RSS AIC
## <none> 4.0829 -45.832
## - X6 1 2.4147 6.4976 -33.893
## - X9 1 2.6501 6.7330 -32.825
## - X1 1 2.6955 6.7784 -32.624
## - X8 1 3.5347 7.6176 -29.122
## - X7 1 5.2580 9.3409 -23.004
## - X3 1 25.3225 29.4054 11.399## Start: AIC=-44.22
## log(y) ~ X1 + X2 + X3 + X4 + X6 + X7 + X8 + X9
##
## Df Sum of Sq RSS AIC
## - X1 1 0.0139 3.7844 -46.110
## - X2 1 0.0279 3.7983 -45.999
## - X4 1 0.1998 3.9703 -44.671
## <none> 3.7705 -44.220
## - X6 1 0.7524 4.5229 -40.762
## - X7 1 1.1605 4.9309 -38.170
## - X3 1 1.6186 5.3891 -35.505
## - X9 1 2.8181 6.5886 -29.476
## - X8 1 3.5442 7.3147 -26.339
##
## Step: AIC=-46.11
## log(y) ~ X2 + X3 + X4 + X6 + X7 + X8 + X9
##
## Df Sum of Sq RSS AIC
## - X2 1 0.0170 3.8013 -47.975
## <none> 3.7844 -46.110
## - X6 1 1.0707 4.8551 -40.635
## - X7 1 1.5504 5.3348 -37.808
## - X9 1 2.8243 6.6087 -31.384
## - X4 1 2.9697 6.7541 -30.731
## - X8 1 3.5305 7.3149 -28.339
## - X3 1 5.3638 9.1482 -21.629
##
## Step: AIC=-47.98
## log(y) ~ X3 + X4 + X6 + X7 + X8 + X9
##
## Df Sum of Sq RSS AIC
## <none> 3.8013 -47.975
## - X6 1 1.0542 4.8555 -42.632
## - X7 1 1.8739 5.6752 -37.953
## - X9 1 2.8256 6.6270 -33.302
## - X4 1 2.9771 6.7784 -32.624
## - X8 1 3.5182 7.3195 -30.320
## - X3 1 5.3653 9.1666 -23.569## Start: AIC=-39.8
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X7 + X8 + X9
##
## Df Sum of Sq RSS AIC
## - X3 1 0.0564 4.4250 -41.418
## - X2 1 0.1043 4.4729 -41.095
## - X7 1 0.1349 4.5035 -40.891
## - X5 1 0.1543 4.5229 -40.762
## <none> 4.3686 -39.803
## - X1 1 0.4338 4.8023 -38.963
## - X4 1 1.7475 6.1160 -31.708
## - X9 1 2.6668 7.0353 -27.508
## - X8 1 3.1935 7.5620 -25.342
##
## Step: AIC=-41.42
## log(y) ~ X1 + X2 + X4 + X5 + X7 + X8 + X9
##
## Df Sum of Sq RSS AIC
## - X7 1 0.0848 4.5098 -42.849
## - X2 1 0.3043 4.7293 -41.423
## <none> 4.4250 -41.418
## - X5 1 0.9642 5.3892 -37.504
## - X9 1 3.3632 7.7882 -26.458
## - X8 1 3.9187 8.3437 -24.391
## - X1 1 4.6412 9.0662 -21.899
## - X4 1 16.0173 20.4423 2.492
##
## Step: AIC=-42.85
## log(y) ~ X1 + X2 + X4 + X5 + X8 + X9
##
## Df Sum of Sq RSS AIC
## <none> 4.5098 -42.849
## - X2 1 0.6110 5.1208 -41.037
## - X5 1 0.8871 5.3969 -39.461
## - X9 1 3.2799 7.7896 -28.452
## - X8 1 3.8347 8.3444 -26.388
## - X1 1 5.0057 9.5155 -22.448
## - X4 1 15.9600 20.4698 0.533## Start: AIC=-39.78
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X8 + X9
##
## Df Sum of Sq RSS AIC
## - X3 1 0.0003 4.3729 -41.773
## - X6 1 0.1309 4.5035 -40.891
## <none> 4.3726 -39.775
## - X5 1 0.5584 4.9309 -38.170
## - X2 1 0.6554 5.0280 -37.585
## - X1 1 0.9371 5.3097 -35.950
## - X9 1 3.0659 7.4385 -25.836
## - X8 1 3.6837 8.0563 -23.442
## - X4 1 3.8737 8.2463 -22.743
##
## Step: AIC=-41.77
## log(y) ~ X1 + X2 + X4 + X5 + X6 + X8 + X9
##
## Df Sum of Sq RSS AIC
## - X6 1 0.1369 4.5098 -42.849
## <none> 4.3729 -41.773
## - X2 1 0.6577 5.0306 -39.570
## - X5 1 1.0236 5.3965 -37.464
## - X9 1 3.4161 7.7890 -26.455
## - X8 1 3.9564 8.3293 -24.442
## - X1 1 4.7933 9.1662 -21.570
## - X4 1 13.8200 18.1929 -1.005
##
## Step: AIC=-42.85
## log(y) ~ X1 + X2 + X4 + X5 + X8 + X9
##
## Df Sum of Sq RSS AIC
## <none> 4.5098 -42.849
## - X2 1 0.6110 5.1208 -41.037
## - X5 1 0.8871 5.3969 -39.461
## - X9 1 3.2799 7.7896 -28.452
## - X8 1 3.8347 8.3444 -26.388
## - X1 1 5.0057 9.5155 -22.448
## - X4 1 15.9600 20.4698 0.533## Start: AIC=-25.2
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X9
##
## Df Sum of Sq RSS AIC
## - X9 1 0.00002 7.1076 -27.201
## - X4 1 0.01076 7.1183 -27.156
## - X2 1 0.05385 7.1614 -26.975
## - X1 1 0.13496 7.2425 -26.637
## - X5 1 0.20709 7.3147 -26.340
## - X6 1 0.45446 7.5620 -25.342
## <none> 7.1076 -25.201
## - X7 1 0.94872 8.0563 -23.442
## - X3 1 1.22165 8.3292 -22.443
##
## Step: AIC=-27.2
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7
##
## Df Sum of Sq RSS AIC
## - X4 1 0.01127 7.1189 -29.154
## - X2 1 0.05390 7.1615 -28.974
## - X1 1 0.13699 7.2446 -28.628
## - X5 1 0.20769 7.3153 -28.337
## - X6 1 0.45911 7.5667 -27.323
## <none> 7.1076 -27.201
## - X7 1 0.95446 8.0621 -25.421
## - X3 1 1.25421 8.3618 -24.326
##
## Step: AIC=-29.15
## log(y) ~ X1 + X2 + X3 + X5 + X6 + X7
##
## Df Sum of Sq RSS AIC
## - X2 1 0.1409 7.2598 -30.5654
## - X5 1 0.4878 7.6066 -29.1654
## <none> 7.1189 -29.1535
## - X6 1 1.1498 8.2686 -26.6619
## - X1 1 2.6634 9.7823 -21.6187
## - X7 1 3.8818 11.0007 -18.0972
## - X3 1 17.0862 24.2051 5.5609
##
## Step: AIC=-30.57
## log(y) ~ X1 + X3 + X5 + X6 + X7
##
## Df Sum of Sq RSS AIC
## - X5 1 0.3665 7.6263 -31.0878
## <none> 7.2598 -30.5654
## - X6 1 1.1128 8.3726 -28.2870
## - X1 1 2.6550 9.9148 -23.2150
## - X7 1 4.5672 11.8270 -17.9245
## - X3 1 19.3750 26.6348 6.4306
##
## Step: AIC=-31.09
## log(y) ~ X1 + X3 + X6 + X7
##
## Df Sum of Sq RSS AIC
## <none> 7.626 -31.088
## - X6 1 1.5290 9.155 -27.606
## - X1 1 2.7319 10.358 -23.902
## - X7 1 4.8252 12.452 -18.381
## - X3 1 26.0905 33.717 11.504## Start: AIC=-28.16
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8
##
## Df Sum of Sq RSS AIC
## - X4 1 0.00125 6.4404 -30.158
## - X2 1 0.03429 6.4735 -30.005
## - X5 1 0.14939 6.5886 -29.476
## - X1 1 0.15258 6.5918 -29.461
## <none> 6.4392 -28.164
## - X6 1 0.59615 7.0353 -27.508
## - X8 1 0.66842 7.1076 -27.201
## - X7 1 0.99933 7.4385 -25.836
## - X3 1 1.34802 7.7872 -24.462
##
## Step: AIC=-30.16
## log(y) ~ X1 + X2 + X3 + X5 + X6 + X7 + X8
##
## Df Sum of Sq RSS AIC
## - X2 1 0.0670 6.5074 -31.848
## - X5 1 0.2880 6.7284 -30.846
## <none> 6.4404 -30.158
## - X8 1 0.6784 7.1189 -29.153
## - X6 1 1.2983 7.7387 -26.649
## - X1 1 2.0749 8.5153 -23.780
## - X7 1 3.5965 10.0369 -18.848
## - X3 1 16.2255 22.6659 5.590
##
## Step: AIC=-31.85
## log(y) ~ X1 + X3 + X5 + X6 + X7 + X8
##
## Df Sum of Sq RSS AIC
## - X5 1 0.2255 6.7330 -32.825
## <none> 6.5074 -31.848
## - X8 1 0.7523 7.2598 -30.565
## - X6 1 1.2782 7.7856 -28.468
## - X1 1 2.1911 8.6986 -25.141
## - X7 1 4.5538 11.0612 -17.933
## - X3 1 18.9765 25.4839 7.105
##
## Step: AIC=-32.83
## log(y) ~ X1 + X3 + X6 + X7 + X8
##
## Df Sum of Sq RSS AIC
## <none> 6.733 -32.825
## - X8 1 0.8934 7.626 -31.088
## - X6 1 1.6647 8.398 -28.197
## - X1 1 2.4940 9.227 -25.372
## - X7 1 4.7585 11.491 -18.788
## - X3 1 26.5284 33.261 13.096# Compare vif
ols_vif_tol(model_wf_full_log)## # A tibble: 9 x 3
## Variables Tolerance VIF
## <chr> <dbl> <dbl>
## 1 X1 0.00982 102.
## 2 X2 0.133 7.52
## 3 X3 0.0318 31.4
## 4 X4 0.00946 106.
## 5 X5 0.103 9.68
## 6 X6 0.433 2.31
## 7 X7 0.0487 20.5
## 8 X8 0.182 5.50
## 9 X9 0.174 5.75ols_vif_tol(model_wf_aic_log)## # A tibble: 6 x 3
## Variables Tolerance VIF
## <chr> <dbl> <dbl>
## 1 X3 0.332 3.01
## 2 X4 0.167 5.97
## 3 X6 0.839 1.19
## 4 X7 0.159 6.28
## 5 X8 0.202 4.94
## 6 X9 0.195 5.12ols_vif_tol(model_wf_rm1_log)## # A tibble: 8 x 3
## Variables Tolerance VIF
## <chr> <dbl> <dbl>
## 1 X2 0.245 4.08
## 2 X3 0.318 3.14
## 3 X4 0.115 8.70
## 4 X5 0.283 3.54
## 5 X6 0.717 1.39
## 6 X7 0.118 8.46
## 7 X8 0.190 5.27
## 8 X9 0.185 5.41ols_vif_tol(model_wf_rm1_aic_log)## # A tibble: 6 x 3
## Variables Tolerance VIF
## <chr> <dbl> <dbl>
## 1 X3 0.332 3.01
## 2 X4 0.167 5.97
## 3 X6 0.839 1.19
## 4 X7 0.159 6.28
## 5 X8 0.202 4.94
## 6 X9 0.195 5.12ols_vif_tol(model_wf_rm2_log)## # A tibble: 8 x 3
## Variables Tolerance VIF
## <chr> <dbl> <dbl>
## 1 X1 0.0181 55.2
## 2 X3 0.0583 17.2
## 3 X4 0.0148 67.4
## 4 X5 0.163 6.13
## 5 X6 0.543 1.84
## 6 X7 0.114 8.79
## 7 X8 0.183 5.46
## 8 X9 0.175 5.72ols_vif_tol(model_wf_rm2_aic_log)## # A tibble: 6 x 3
## Variables Tolerance VIF
## <chr> <dbl> <dbl>
## 1 X3 0.332 3.01
## 2 X4 0.167 5.97
## 3 X6 0.839 1.19
## 4 X7 0.159 6.28
## 5 X8 0.202 4.94
## 6 X9 0.195 5.12ols_vif_tol(model_wf_rm3_log)## # A tibble: 8 x 3
## Variables Tolerance VIF
## <chr> <dbl> <dbl>
## 1 X1 0.0983 10.2
## 2 X2 0.243 4.11
## 3 X4 0.113 8.87
## 4 X5 0.272 3.68
## 5 X6 0.767 1.30
## 6 X7 0.206 4.85
## 7 X8 0.190 5.26
## 8 X9 0.187 5.36ols_vif_tol(model_wf_rm3_aic_log)## # A tibble: 6 x 3
## Variables Tolerance VIF
## <chr> <dbl> <dbl>
## 1 X1 0.119 8.39
## 2 X2 0.313 3.19
## 3 X4 0.123 8.12
## 4 X5 0.343 2.92
## 5 X8 0.209 4.79
## 6 X9 0.205 4.88ols_vif_tol(model_wf_rm4_log)## # A tibble: 8 x 3
## Variables Tolerance VIF
## <chr> <dbl> <dbl>
## 1 X1 0.119 8.38
## 2 X2 0.209 4.79
## 3 X3 0.379 2.64
## 4 X5 0.187 5.35
## 5 X6 0.836 1.20
## 6 X7 0.165 6.05
## 7 X8 0.187 5.35
## 8 X9 0.183 5.45ols_vif_tol(model_wf_rm4_aic_log)## # A tibble: 6 x 3
## Variables Tolerance VIF
## <chr> <dbl> <dbl>
## 1 X1 0.279 3.58
## 2 X3 0.768 1.30
## 3 X6 0.917 1.09
## 4 X7 0.251 3.99
## 5 X8 0.202 4.94
## 6 X9 0.196 5.10ols_vif_tol(model_wf_rm5_log)## # A tibble: 8 x 3
## Variables Tolerance VIF
## <chr> <dbl> <dbl>
## 1 X1 0.0269 37.2
## 2 X2 0.210 4.77
## 3 X3 0.0836 12.0
## 4 X4 0.0171 58.4
## 5 X6 0.485 2.06
## 6 X7 0.0774 12.9
## 7 X8 0.200 5.01
## 8 X9 0.190 5.25ols_vif_tol(model_wf_rm5_aic_log)## # A tibble: 6 x 3
## Variables Tolerance VIF
## <chr> <dbl> <dbl>
## 1 X3 0.332 3.01
## 2 X4 0.167 5.97
## 3 X6 0.839 1.19
## 4 X7 0.159 6.28
## 5 X8 0.202 4.94
## 6 X9 0.195 5.12ols_vif_tol(model_wf_rm6_log)## # A tibble: 8 x 3
## Variables Tolerance VIF
## <chr> <dbl> <dbl>
## 1 X1 0.0162 61.5
## 2 X2 0.167 6.00
## 3 X3 0.0563 17.8
## 4 X4 0.0182 54.8
## 5 X5 0.116 8.64
## 6 X7 0.0832 12.0
## 7 X8 0.182 5.49
## 8 X9 0.174 5.75ols_vif_tol(model_wf_rm6_aic_log)## # A tibble: 6 x 3
## Variables Tolerance VIF
## <chr> <dbl> <dbl>
## 1 X1 0.119 8.39
## 2 X2 0.313 3.19
## 3 X4 0.123 8.12
## 4 X5 0.343 2.92
## 5 X8 0.209 4.79
## 6 X9 0.205 4.88ols_vif_tol(model_wf_rm7_log)## # A tibble: 8 x 3
## Variables Tolerance VIF
## <chr> <dbl> <dbl>
## 1 X1 0.0238 42.0
## 2 X2 0.310 3.22
## 3 X3 0.135 7.42
## 4 X4 0.0321 31.1
## 5 X5 0.164 6.09
## 6 X6 0.740 1.35
## 7 X8 0.184 5.45
## 8 X9 0.177 5.66ols_vif_tol(model_wf_rm7_aic_log)## # A tibble: 6 x 3
## Variables Tolerance VIF
## <chr> <dbl> <dbl>
## 1 X1 0.119 8.39
## 2 X2 0.313 3.19
## 3 X4 0.123 8.12
## 4 X5 0.343 2.92
## 5 X8 0.209 4.79
## 6 X9 0.205 4.88ols_vif_tol(model_wf_rm8_log)## # A tibble: 8 x 3
## Variables Tolerance VIF
## <chr> <dbl> <dbl>
## 1 X1 0.0103 97.5
## 2 X2 0.134 7.46
## 3 X3 0.0333 30.0
## 4 X4 0.00973 103.
## 5 X5 0.114 8.81
## 6 X6 0.435 2.30
## 7 X7 0.0492 20.3
## 8 X9 0.879 1.14ols_vif_tol(model_wf_rm8_aic_log)## # A tibble: 4 x 3
## Variables Tolerance VIF
## <chr> <dbl> <dbl>
## 1 X1 0.281 3.56
## 2 X3 0.773 1.29
## 3 X6 0.949 1.05
## 4 X7 0.252 3.96ols_vif_tol(model_wf_rm9_log)## # A tibble: 8 x 3
## Variables Tolerance VIF
## <chr> <dbl> <dbl>
## 1 X1 0.0104 95.8
## 2 X2 0.134 7.48
## 3 X3 0.0341 29.3
## 4 X4 0.00998 100.
## 5 X5 0.113 8.83
## 6 X6 0.433 2.31
## 7 X7 0.0495 20.2
## 8 X8 0.919 1.09ols_vif_tol(model_wf_rm9_aic_log)## # A tibble: 5 x 3
## Variables Tolerance VIF
## <chr> <dbl> <dbl>
## 1 X1 0.280 3.58
## 2 X3 0.771 1.30
## 3 X6 0.945 1.06
## 4 X7 0.252 3.97
## 5 X8 0.963 1.04(d) (3) Variable selection
library(huxtable)
huxreg(model_wf_rm1_log, model_wf_rm2_log, model_wf_rm3_log, model_wf_rm4_log, model_wf_rm5_log, model_wf_rm6_log, model_wf_rm7_log, model_wf_rm8_log, model_wf_rm9_log, model_wf_full_log)| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | |
| (Intercept) | 3.280 | 3.703 | 7.731 *** | 1.200 | 2.581 *** | 5.523 | 7.487 ** | -0.182 | -0.127 | 3.402 |
| (1.690) | (2.333) | (1.678) | (2.110) | (0.547) | (3.075) | (2.314) | (4.072) | (3.844) | (3.150) | |
| X2 | -1.243 | 6.526 | -5.001 | -2.144 | 4.655 | 8.550 | -3.730 | -2.980 | -1.024 | |
| (5.025) | (5.358) | (5.568) | (5.445) | (6.574) | (4.819) | (9.350) | (8.912) | (6.995) | ||
| X3 | 0.183 *** | 0.167 * | 0.278 *** | 0.201 ** | 0.046 | 0.002 | 0.277 | 0.287 * | 0.178 | |
| (0.034) | (0.080) | (0.032) | (0.067) | (0.088) | (0.057) | (0.146) | (0.137) | (0.111) | ||
| X4 | 0.104 ** | 0.119 | 0.286 *** | 0.088 | 0.253 ** | 0.284 *** | 0.027 | 0.009 | 0.109 | |
| (0.032) | (0.090) | (0.035) | (0.084) | (0.087) | (0.066) | (0.153) | (0.143) | (0.115) | ||
| X5 | -0.008 | -0.013 | -0.055 * | 0.013 | -0.031 | -0.050 | 0.036 | 0.031 | -0.010 | |
| (0.021) | (0.028) | (0.023) | (0.027) | (0.036) | (0.030) | (0.047) | (0.044) | (0.036) | ||
| X6 | -0.396 * | -0.375 | -0.161 | -0.531 ** | -0.408 | -0.138 | -0.335 | -0.385 | -0.389 | |
| (0.164) | (0.188) | (0.168) | (0.155) | (0.199) | (0.174) | (0.289) | (0.276) | (0.216) | ||
| X7 | 4.317 ** | 3.975 * | 0.959 | 6.088 *** | 4.611 * | 1.517 | 5.230 | 5.352 | 4.233 | |
| (1.466) | (1.495) | (1.178) | (1.266) | (1.814) | (1.884) | (3.124) | (2.965) | (2.339) | ||
| X8 | 0.629 *** | 0.632 *** | 0.680 *** | 0.606 *** | 0.618 *** | 0.614 *** | 0.657 *** | 0.125 | 0.630 *** | |
| (0.142) | (0.145) | (0.151) | (0.147) | (0.139) | (0.157) | (0.156) | (0.085) | (0.149) | ||
| X9 | -0.461 *** | -0.464 *** | -0.513 *** | -0.436 ** | -0.453 *** | -0.461 ** | -0.490 *** | 0.001 | -0.462 ** | |
| (0.116) | (0.119) | (0.122) | (0.119) | (0.114) | (0.129) | (0.128) | (0.073) | (0.122) | ||
| X1 | -0.042 | -0.457 *** | 0.250 ** | 0.048 | -0.345 | -0.418 * | 0.242 | 0.255 | -0.014 | |
| (0.210) | (0.096) | (0.083) | (0.173) | (0.239) | (0.197) | (0.383) | (0.362) | (0.292) | ||
| N | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 |
| R2 | 0.947 | 0.947 | 0.941 | 0.945 | 0.947 | 0.939 | 0.939 | 0.900 | 0.910 | 0.947 |
| logLik | -11.407 | -11.422 | -13.216 | -12.059 | -11.458 | -13.667 | -13.681 | -20.968 | -19.486 | -11.406 |
| AIC | 42.815 | 42.843 | 46.432 | 44.118 | 42.916 | 47.333 | 47.361 | 61.935 | 58.972 | 44.811 |
| *** p < 0.001; ** p < 0.01; * p < 0.05. | ||||||||||
huxreg(model_wf_rm1_aic_log, model_wf_rm2_aic_log, model_wf_rm3_aic_log, model_wf_rm4_aic_log, model_wf_rm5_aic_log, model_wf_rm6_aic_log, model_wf_rm7_aic_log, model_wf_rm8_aic_log, model_wf_rm9_aic_log, model_wf_aic_log)| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | |
| (Intercept) | 2.692 *** | 2.692 *** | 6.882 *** | 2.307 *** | 2.692 *** | 6.882 *** | 6.882 *** | 2.587 *** | 2.225 *** | 2.692 *** |
| (0.445) | (0.445) | (1.432) | (0.410) | (0.445) | (1.432) | (1.432) | (0.494) | (0.515) | (0.445) | |
| X3 | 0.184 *** | 0.184 *** | 0.263 *** | 0.184 *** | 0.266 *** | 0.268 *** | 0.184 *** | |||
| (0.032) | (0.032) | (0.022) | (0.032) | (0.029) | (0.028) | (0.032) | ||||
| X4 | 0.109 *** | 0.109 *** | 0.294 *** | 0.109 *** | 0.294 *** | 0.294 *** | 0.109 *** | |||
| (0.026) | (0.026) | (0.033) | (0.026) | (0.033) | (0.033) | (0.026) | ||||
| X6 | -0.368 * | -0.368 * | -0.532 ** | -0.368 * | -0.416 * | -0.435 * | -0.368 * | |||
| (0.146) | (0.146) | (0.144) | (0.146) | (0.186) | (0.179) | (0.146) | ||||
| X7 | 4.085 ** | 4.085 ** | 5.453 *** | 4.085 ** | 5.209 *** | 5.174 *** | 4.085 ** | |||
| (1.213) | (1.213) | (1.002) | (1.213) | (1.310) | (1.256) | (1.213) | ||||
| X8 | 0.612 *** | 0.612 *** | 0.629 *** | 0.613 *** | 0.612 *** | 0.629 *** | 0.629 *** | 0.141 | 0.612 *** | |
| (0.133) | (0.133) | (0.142) | (0.137) | (0.133) | (0.142) | (0.142) | (0.079) | (0.133) | ||
| X9 | -0.448 *** | -0.448 *** | -0.471 *** | -0.433 *** | -0.448 *** | -0.471 *** | -0.471 *** | -0.448 *** | ||
| (0.108) | (0.108) | (0.115) | (0.112) | (0.108) | (0.115) | (0.115) | (0.108) | |||
| X1 | -0.432 *** | 0.207 *** | -0.432 *** | -0.432 *** | 0.208 ** | 0.199 ** | ||||
| (0.086) | (0.053) | (0.086) | (0.086) | (0.070) | (0.067) | |||||
| X2 | 8.217 | 8.217 | 8.217 | |||||||
| (4.655) | (4.655) | (4.655) | ||||||||
| X5 | -0.043 * | -0.043 * | -0.043 * | |||||||
| (0.020) | (0.020) | (0.020) | ||||||||
| N | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 |
| R2 | 0.947 | 0.947 | 0.937 | 0.943 | 0.947 | 0.937 | 0.937 | 0.893 | 0.906 | 0.947 |
| logLik | -11.581 | -11.581 | -14.144 | -12.652 | -11.581 | -14.144 | -14.144 | -22.024 | -20.155 | -11.581 |
| AIC | 39.161 | 39.161 | 44.288 | 41.305 | 39.161 | 44.288 | 44.288 | 56.049 | 54.311 | 39.161 |
| *** p < 0.001; ** p < 0.01; * p < 0.05. | ||||||||||
(d) (4) Forward selection
Stepwise Forward Regression for full model
# Stepwise Forward Regression based on p values (use a=0.15) #
ols_step_forward_p(model_wf_full_log, penter = 0.15)
# Stepwise AIC Forward Regression #
ols_step_forward_aic(model_wf_full_log)Stepwise Forward Regression for X4 eliminated model
# Stepwise Forward Regression based on p values (use a=0.15) #
ols_step_forward_p(model_wf_rm4_log, penter = 0.15)
# Stepwise AIC Forward Regression #
ols_step_forward_aic(model_wf_rm4_log)Stepwise Forward Regression for X1 eliminated model
# Stepwise Forward Regression based on p values (use a=0.15) #
ols_step_forward_p(model_wf_rm1_log, penter = 0.15)
# Stepwise AIC Forward Regression #
ols_step_forward_aic(model_wf_rm1_log)(d) (5) Backward selection
Stepwise Backward Regression for full model
# Stepwise Backward Regression based on p values (use a=0.05) #
ols_step_backward_p(model_wf_full_log, penter = 0.05)## Backward Elimination Method
## ---------------------------
##
## Candidate Terms:
##
## 1 . X1
## 2 . X2
## 3 . X3
## 4 . X4
## 5 . X5
## 6 . X6
## 7 . X7
## 8 . X8
## 9 . X9
##
## We are eliminating variables based on p value...
##
## Variables Removed:
##
## - X1
## - X2
## - X5
##
## No more variables satisfy the condition of p value = 0.3
##
##
## Final Model Output
## ------------------
##
## Model Summary
## -------------------------------------------------------------
## R 0.973 RMSE 0.407
## R-Squared 0.947 Coef. Var 6.385
## Adj. R-Squared 0.933 MSE 0.165
## Pred R-Squared 0.908 MAE 0.273
## -------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## -------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## -------------------------------------------------------------------
## Regression 67.591 6 11.265 68.16 0.0000
## Residual 3.801 23 0.165
## Total 71.393 29
## -------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) 2.692 0.445 6.046 0.000 1.771 3.613
## X3 0.184 0.032 0.476 5.698 0.000 0.117 0.251
## X4 0.109 0.026 0.499 4.244 0.000 0.056 0.162
## X6 -0.368 0.146 -0.133 -2.526 0.019 -0.669 -0.066
## X7 4.085 1.213 0.406 3.367 0.003 1.575 6.595
## X8 0.612 0.133 0.493 4.614 0.000 0.337 0.886
## X9 -0.448 0.108 -0.450 -4.135 0.000 -0.672 -0.224
## ----------------------------------------------------------------------------------------##
##
## Elimination Summary
## -----------------------------------------------------------------------
## Variable Adj.
## Step Removed R-Square R-Square C(p) AIC RMSE
## -----------------------------------------------------------------------
## 1 X1 0.9474 0.9273 8.0021 42.8146 0.4230
## 2 X2 0.9472 0.9304 6.0604 40.9019 0.4139
## 3 X5 0.9468 0.9329 4.2345 39.1611 0.4065
## -----------------------------------------------------------------------# Stepwise AIC Backward Regression #
ols_step_backward_aic(model_wf_full_log)## Backward Elimination Method
## ---------------------------
##
## Candidate Terms:
##
## 1 . X1
## 2 . X2
## 3 . X3
## 4 . X4
## 5 . X5
## 6 . X6
## 7 . X7
## 8 . X8
## 9 . X9
##
##
## Variables Removed:
##
## - X1
## - X2
## - X5
##
## No more variables to be removed.##
##
## Backward Elimination Summary
## ---------------------------------------------------------------
## Variable AIC RSS Sum Sq R-Sq Adj. R-Sq
## ---------------------------------------------------------------
## Full Model 44.811 3.757 67.635 0.94737 0.92369
## X1 42.815 3.758 67.635 0.94737 0.92731
## X2 40.902 3.769 67.624 0.94721 0.93042
## X5 39.161 3.801 67.591 0.94675 0.93286
## ---------------------------------------------------------------Stepwise Backward Regression for X4 eliminated model
# Stepwise Backward Regression based on p values (use a=0.05) #
ols_step_backward_p(model_wf_rm4_log, penter = 0.05)
# Stepwise AIC Backward Regression #
ols_step_backward_aic(model_wf_rm4_log)Stepwise Backward Regression for X1 eliminated model
# Stepwise Backward Regression based on p values (use a=0.05) #
ols_step_backward_p(model_wf_rm1_log, penter = 0.05)
# Stepwise AIC Backward Regression #
ols_step_backward_aic(model_wf_rm1_log)(d) (6) Best Subset Regression
# For full model #
k <- ols_step_best_subset(model_wf_full_log)
k| mindex | n | predictors | rsquare | adjr | predrsq | cp | aic | sbic | sbc | msep | fpe | apc | hsp |
| 1 | 1 | X4 | 0.803 | 0.796 | 0.772 | 48.9 | 68.4 | -19.8 | 72.6 | 0.538 | 0.536 | 0.225 | 0.0186 |
| 2 | 2 | X3 X4 | 0.873 | 0.864 | 0.844 | 24.2 | 57.2 | -30.5 | 62.8 | 0.373 | 0.369 | 0.155 | 0.0129 |
| 3 | 3 | X3 X4 X7 | 0.89 | 0.878 | 0.854 | 19.7 | 54.8 | -32.7 | 61.8 | 0.348 | 0.341 | 0.143 | 0.012 |
| 4 | 4 | X1 X4 X8 X9 | 0.921 | 0.908 | 0.886 | 10.1 | 47 | -38.1 | 55.4 | 0.272 | 0.264 | 0.111 | 0.00941 |
| 5 | 5 | X3 X4 X7 X8 X9 | 0.932 | 0.918 | 0.892 | 7.85 | 44.5 | -38.8 | 54.3 | 0.255 | 0.243 | 0.102 | 0.0088 |
| 6 | 6 | X3 X4 X6 X7 X8 X9 | 0.947 | 0.933 | 0.908 | 4.23 | 39.2 | -39.7 | 50.4 | 0.217 | 0.204 | 0.0857 | 0.00751 |
| 7 | 7 | X3 X4 X5 X6 X7 X8 X9 | 0.947 | 0.93 | 0.902 | 6.06 | 40.9 | -36.8 | 53.5 | 0.236 | 0.217 | 0.0912 | 0.00816 |
| 8 | 8 | X2 X3 X4 X5 X6 X7 X8 X9 | 0.947 | 0.927 | 0.896 | 8 | 42.8 | -33.8 | 56.8 | 0.259 | 0.233 | 0.0977 | 0.00895 |
| 9 | 9 | X1 X2 X3 X4 X5 X6 X7 X8 X9 | 0.947 | 0.924 | 0.886 | 10 | 44.8 | -30.8 | 60.2 | 0.286 | 0.25 | 0.105 | 0.00989 |
plot(k)
# For X4 eliminated model #
# k <- ols_step_best_subset(model_wf_rm4_log)
# k
# plot(k)
# For X1 eliminated model #
# k <- ols_step_best_subset(model_wf_rm1_log)
# k
# plot(k)(d) (7) Models Comparison
- Model 437896
# build model 437896
model_wf_437896_log <- lm(log(y) ~ X4 + X3 + X7 + X8 + X9 + X6, data=table_wf)
ols_regress(model_wf_437896_log)## Model Summary
## -------------------------------------------------------------
## R 0.973 RMSE 0.407
## R-Squared 0.947 Coef. Var 6.385
## Adj. R-Squared 0.933 MSE 0.165
## Pred R-Squared 0.908 MAE 0.273
## -------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## -------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## -------------------------------------------------------------------
## Regression 67.591 6 11.265 68.16 0.0000
## Residual 3.801 23 0.165
## Total 71.393 29
## -------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) 2.692 0.445 6.046 0.000 1.771 3.613
## X4 0.109 0.026 0.499 4.244 0.000 0.056 0.162
## X3 0.184 0.032 0.476 5.698 0.000 0.117 0.251
## X7 4.085 1.213 0.406 3.367 0.003 1.575 6.595
## X8 0.612 0.133 0.493 4.614 0.000 0.337 0.886
## X9 -0.448 0.108 -0.450 -4.135 0.000 -0.672 -0.224
## X6 -0.368 0.146 -0.133 -2.526 0.019 -0.669 -0.066
## ----------------------------------------------------------------------------------------confint(model_wf_437896_log, level=0.95/1) # Bonferroni joint confidence interval ### 2.5 % 97.5 %
## (Intercept) 1.77080533 3.61278901
## X4 0.05589763 0.16220302
## X3 0.11709482 0.25059326
## X7 1.57533753 6.59460252
## X8 0.33738118 0.88582991
## X9 -0.67159465 -0.22368135
## X6 -0.66855123 -0.06648201# Collinearity Diagnostics #
ols_vif_tol(model_wf_437896_log)| Variables | Tolerance | VIF |
| X4 | 0.167 | 5.97 |
| X3 | 0.332 | 3.01 |
| X7 | 0.159 | 6.28 |
| X8 | 0.202 | 4.94 |
| X9 | 0.195 | 5.12 |
| X6 | 0.839 | 1.19 |
#Model Fit Assessment
ols_plot_diagnostics(model_wf_437896_log)
# Part & Partial Correlations
ols_test_correlation(model_wf_437896_log) # Correlation between observed residuals and expected residuals under normality.## [1] 0.9837263# Residual Normality Test
ols_test_normality(model_wf_437896_log) # Test for detecting violation of normality assumption. #If p-value is bigger, then no problem of non-normality ### -----------------------------------------------
## Test Statistic pvalue
## -----------------------------------------------
## Shapiro-Wilk 0.9728 0.6175
## Kolmogorov-Smirnov 0.0997 0.8982
## Cramer-von Mises 4.8429 0.0000
## Anderson-Darling 0.2996 0.5612
## -----------------------------------------------# Variable Contributions
ols_plot_added_variable(model_wf_437896_log)
# Residual Plus Component Plot
ols_plot_comp_plus_resid(model_wf_437896_log)
- Model 437
# build model 437
model_wf_437_log <- lm(log(y) ~ X4 + X3 + X7, data=table_wf)
ols_regress(model_wf_437_log)## Model Summary
## -------------------------------------------------------------
## R 0.944 RMSE 0.549
## R-Squared 0.890 Coef. Var 8.618
## Adj. R-Squared 0.878 MSE 0.301
## Pred R-Squared 0.854 MAE 0.414
## -------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## -------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## -------------------------------------------------------------------
## Regression 63.565 3 21.188 70.378 0.0000
## Residual 7.828 26 0.301
## Total 71.393 29
## -------------------------------------------------------------------
##
## Parameter Estimates
## -------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## -------------------------------------------------------------------------------------
## (Intercept) 2.872 0.547 5.254 0.000 1.748 3.995
## X4 0.122 0.033 0.559 3.730 0.001 0.055 0.189
## X3 0.168 0.040 0.435 4.165 0.000 0.085 0.251
## X7 3.106 1.537 0.309 2.021 0.054 -0.053 6.266
## -------------------------------------------------------------------------------------# Collinearity Diagnostics #
ols_vif_tol(model_wf_437_log)| Variables | Tolerance | VIF |
| X4 | 0.188 | 5.32 |
| X3 | 0.386 | 2.59 |
| X7 | 0.181 | 5.53 |
#Model Fit Assessment
ols_plot_diagnostics(model_wf_437_log)
# Part & Partial Correlations
ols_test_correlation(model_wf_437_log) # Correlation between observed residuals and expected residuals under normality.## [1] 0.9856766# Residual Normality Test
ols_test_normality(model_wf_437_log) # Test for detecting violation of normality assumption. #If p-value is bigger, then no problem of non-normality ### -----------------------------------------------
## Test Statistic pvalue
## -----------------------------------------------
## Shapiro-Wilk 0.9765 0.7267
## Kolmogorov-Smirnov 0.1033 0.8736
## Cramer-von Mises 3.1908 0.0000
## Anderson-Darling 0.3511 0.4469
## -----------------------------------------------# Variable Contributions
ols_plot_added_variable(model_wf_437_log)
# Residual Plus Component Plot
ols_plot_comp_plus_resid(model_wf_437_log)
- Model 137689
# build model 137689
model_wf_137689_log <- lm(log(y) ~ X1 + X3 + X7 + X6 + X8 + X9, data=table_wf)
ols_regress(model_wf_137689_log)## Model Summary
## -------------------------------------------------------------
## R 0.971 RMSE 0.421
## R-Squared 0.943 Coef. Var 6.618
## Adj. R-Squared 0.928 MSE 0.178
## Pred R-Squared 0.900 MAE 0.292
## -------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## -------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## -------------------------------------------------------------------
## Regression 67.310 6 11.218 63.195 0.0000
## Residual 4.083 23 0.178
## Total 71.393 29
## -------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) 2.307 0.410 5.623 0.000 1.458 3.156
## X1 0.207 0.053 0.368 3.897 0.001 0.097 0.317
## X3 0.263 0.022 0.680 11.944 0.000 0.217 0.308
## X7 5.453 1.002 0.542 5.442 0.000 3.380 7.525
## X6 -0.532 0.144 -0.192 -3.688 0.001 -0.831 -0.234
## X8 0.613 0.137 0.495 4.462 0.000 0.329 0.897
## X9 -0.433 0.112 -0.435 -3.864 0.001 -0.665 -0.201
## ----------------------------------------------------------------------------------------# Collinearity Diagnostics #
ols_vif_tol(model_wf_137689_log)| Variables | Tolerance | VIF |
| X1 | 0.279 | 3.58 |
| X3 | 0.768 | 1.3 |
| X7 | 0.251 | 3.99 |
| X6 | 0.917 | 1.09 |
| X8 | 0.202 | 4.94 |
| X9 | 0.196 | 5.1 |
#Model Fit Assessment
ols_plot_diagnostics(model_wf_137689_log)
# Part & Partial Correlations
ols_test_correlation(model_wf_137689_log) # Correlation between observed residuals and expected residuals under normality.## [1] 0.988106# Residual Normality Test
ols_test_normality(model_wf_137689_log) # Test for detecting violation of normality assumption. #If p-value is bigger, then no problem of non-normality ### -----------------------------------------------
## Test Statistic pvalue
## -----------------------------------------------
## Shapiro-Wilk 0.9769 0.7382
## Kolmogorov-Smirnov 0.0771 0.9881
## Cramer-von Mises 4.4689 0.0000
## Anderson-Darling 0.1644 0.9350
## -----------------------------------------------# Variable Contributions
ols_plot_added_variable(model_wf_137689_log)
# Residual Plus Component Plot
ols_plot_comp_plus_resid(model_wf_137689_log)
# Check PRESS Statistic
ols_press(model_wf_full)## [1] 15880486ols_press(model_wf_full_log)## [1] 8.136733ols_press(model_wf_437896_log)## [1] 6.538275ols_press(model_wf_437_log)## [1] 10.43262ols_press(model_wf_137689_log)## [1] 7.114336# prediction power
ols_pred_rsq(model_wf_full)## [1] 0.6179649ols_pred_rsq(model_wf_full_log)## [1] 0.8860283ols_pred_rsq(model_wf_437896_log)## [1] 0.908418ols_pred_rsq(model_wf_437_log)## [1] 0.8538697ols_pred_rsq(model_wf_137689_log)## [1] 0.900349More
- Other Models
# build X1*X8 eliminated log model
model_wf_18rm4_log <- lm(log(y) ~ X1*X8 + X3 + X6 + X7 + X9, data=table_wf)
# build X1*X8 eliminated log model
table_wf_resi <- table_wf%>% mutate(x1t8=X1*X8)
model_wf_1time8_log <- lm(log(y) ~ x1t8 + X3 + X6 + X7+ X9 , data=table_wf_resi)
# build X1*X4 eliminated log model
table_wf_resi <- table_wf%>% mutate(x1t4=X1*X4)
model_wf_1time4_log <- lm(log(y) ~ x1t4 + X3 + X6 + X7+ X8+ X9, data=table_wf_resi)
summary(model_wf_1time4_log)
# build X1/X4 eliminated log model
table_wf_resi <- table_wf%>% mutate(x14=X1/X4)
model_wf_1per4_log <- lm(log(y) ~ x14 + X3 + X6 + X7+ X8+ X9, data=table_wf_resi)
# build X4*X3 eliminated log model
model_wf_43rm1_log <- lm(log(y) ~ X9 + X4*X3 + X6 + X7 + X8 , data=table_wf)
# build X4*X9 eliminated log model
model_wf_49rm1_log <- lm(log(y) ~ X3 + X4*X9 + X6 + X7 + X8 , data=table_wf)
# build X4*X9 eliminated log model
model_wf_48rm1_log <- lm(log(y) ~ X3 + X4*X8 + X6 + X7 + X9 , data=table_wf)
# build X4*X9 eliminated log model
model_wf_47rm1_log <- lm(log(y) ~ X3 + X4*X7 + X6 + X9 + X8 , data=table_wf)
# build X4/X9 eliminated log model
table_wf_resi <- table_wf%>% mutate(x4p9=X4/X9)
model_wf_4per9_log <- lm(log(y) ~ X3 + x4p9 + X6 + X7 + X8 , data=table_wf_resi)
# build X3/X4vX8*X9 eliminated log model
model_wf_34v89_log <- lm(log(y) ~ X3*X4 + X8*X9 + X6 + X7, data=table_wf_resi)
# build X3/X4vX8*X9 eliminated log model
model_wf_34v89v67_log <- lm(log(y) ~ X3*X4 + X8*X9 + X6*X7, data=table_wf_resi)
# build X8/X9vX4*X3 eliminated log model
table_wf_resi <- table_wf%>% mutate(x8p9=X8/X9)
model_wf_8per9v43_log <- lm(log(y) ~ x8p9 + X4*X3 + X6 + X7, data=table_wf_resi)
# build X6/7vX8/X9vX4X3 eliminated log model
table_wf_resi <- table_wf%>% mutate(x8p9=X8/X9,x6p7=X6/X7)
model_wf_6p7v8p9v43_log <- lm(log(y) ~ x8p9 + X4*X3 + x6p7, data=table_wf_resi)
# build X8/X9vX4*X3rmX7 eliminated log model
table_wf_resi <- table_wf%>% mutate(x8p9=X8/X9)
model_wf_8per9v43rm7_log <- lm(log(y) ~ x8p9 + X4*X3 + X6, data=table_wf_resi)
# build X8/X9vX4*X3vX6/X7 eliminated log model
table_wf_resi <- table_wf%>% mutate(x8p9=X8/X9)
model_wf_8per9v43rm7_log <- lm(log(y) ~ x8p9 + X4*X3 + X6, data=table_wf_resi)
huxreg(model_wf_8per9v43rm7_log, model_wf_8per9v43_log, model_wf_43rm1_log, model_wf_6p7v8p9v43_log, model_wf_34v89_log, model_wf_34v89v67_log)- Interaction models
# Interaction regression for full
model_wf_full_log_inter <- lm(log(y)~ (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9)^2, data=table_wf)
model_wf_aic_log_inter <- stepAIC(model_wf_full_log_inter)## Start: AIC=-86.68
## log(y) ~ (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9)^2
##
##
## Step: AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 +
## X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X7 + X1:X8 + X1:X9 + X2:X3 +
## X2:X4 + X2:X5 + X2:X6 + X2:X7 + X2:X8 + X2:X9 + X3:X4 + X3:X5 +
## X3:X6 + X3:X7 + X3:X8 + X3:X9 + X4:X5 + X4:X6 + X4:X7 + X4:X8 +
## X4:X9 + X5:X6 + X5:X7 + X5:X8 + X5:X9 + X6:X8 + X6:X9 + X7:X8 +
## X7:X9 + X8:X9
##
##
## Step: AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 +
## X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X7 + X1:X8 + X1:X9 + X2:X3 +
## X2:X4 + X2:X5 + X2:X6 + X2:X7 + X2:X8 + X2:X9 + X3:X4 + X3:X5 +
## X3:X6 + X3:X7 + X3:X8 + X3:X9 + X4:X5 + X4:X6 + X4:X7 + X4:X8 +
## X4:X9 + X5:X6 + X5:X8 + X5:X9 + X6:X8 + X6:X9 + X7:X8 + X7:X9 +
## X8:X9
##
##
## Step: AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 +
## X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X7 + X1:X8 + X1:X9 + X2:X3 +
## X2:X4 + X2:X5 + X2:X6 + X2:X7 + X2:X8 + X2:X9 + X3:X4 + X3:X5 +
## X3:X6 + X3:X7 + X3:X8 + X3:X9 + X4:X5 + X4:X6 + X4:X7 + X4:X8 +
## X4:X9 + X5:X8 + X5:X9 + X6:X8 + X6:X9 + X7:X8 + X7:X9 + X8:X9
##
##
## Step: AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 +
## X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X7 + X1:X8 + X1:X9 + X2:X3 +
## X2:X4 + X2:X5 + X2:X6 + X2:X7 + X2:X8 + X2:X9 + X3:X4 + X3:X5 +
## X3:X6 + X3:X7 + X3:X8 + X3:X9 + X4:X5 + X4:X6 + X4:X8 + X4:X9 +
## X5:X8 + X5:X9 + X6:X8 + X6:X9 + X7:X8 + X7:X9 + X8:X9
##
##
## Step: AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 +
## X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X7 + X1:X8 + X1:X9 + X2:X3 +
## X2:X4 + X2:X5 + X2:X6 + X2:X7 + X2:X8 + X2:X9 + X3:X4 + X3:X5 +
## X3:X6 + X3:X7 + X3:X8 + X3:X9 + X4:X5 + X4:X8 + X4:X9 + X5:X8 +
## X5:X9 + X6:X8 + X6:X9 + X7:X8 + X7:X9 + X8:X9
##
##
## Step: AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 +
## X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X7 + X1:X8 + X1:X9 + X2:X3 +
## X2:X4 + X2:X5 + X2:X6 + X2:X7 + X2:X8 + X2:X9 + X3:X4 + X3:X5 +
## X3:X6 + X3:X7 + X3:X8 + X3:X9 + X4:X8 + X4:X9 + X5:X8 + X5:X9 +
## X6:X8 + X6:X9 + X7:X8 + X7:X9 + X8:X9
##
##
## Step: AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 +
## X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X7 + X1:X8 + X1:X9 + X2:X3 +
## X2:X4 + X2:X5 + X2:X6 + X2:X7 + X2:X8 + X2:X9 + X3:X4 + X3:X5 +
## X3:X6 + X3:X8 + X3:X9 + X4:X8 + X4:X9 + X5:X8 + X5:X9 + X6:X8 +
## X6:X9 + X7:X8 + X7:X9 + X8:X9
##
##
## Step: AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 +
## X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X7 + X1:X8 + X1:X9 + X2:X3 +
## X2:X4 + X2:X5 + X2:X6 + X2:X7 + X2:X8 + X2:X9 + X3:X4 + X3:X5 +
## X3:X8 + X3:X9 + X4:X8 + X4:X9 + X5:X8 + X5:X9 + X6:X8 + X6:X9 +
## X7:X8 + X7:X9 + X8:X9
##
##
## Step: AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 +
## X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X7 + X1:X8 + X1:X9 + X2:X3 +
## X2:X4 + X2:X5 + X2:X6 + X2:X7 + X2:X8 + X2:X9 + X3:X4 + X3:X8 +
## X3:X9 + X4:X8 + X4:X9 + X5:X8 + X5:X9 + X6:X8 + X6:X9 + X7:X8 +
## X7:X9 + X8:X9
##
##
## Step: AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 +
## X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X7 + X1:X8 + X1:X9 + X2:X3 +
## X2:X4 + X2:X5 + X2:X6 + X2:X7 + X2:X8 + X2:X9 + X3:X8 + X3:X9 +
## X4:X8 + X4:X9 + X5:X8 + X5:X9 + X6:X8 + X6:X9 + X7:X8 + X7:X9 +
## X8:X9
##
##
## Step: AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 +
## X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X7 + X1:X8 + X1:X9 + X2:X3 +
## X2:X4 + X2:X5 + X2:X6 + X2:X8 + X2:X9 + X3:X8 + X3:X9 + X4:X8 +
## X4:X9 + X5:X8 + X5:X9 + X6:X8 + X6:X9 + X7:X8 + X7:X9 + X8:X9
##
##
## Step: AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 +
## X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X7 + X1:X8 + X1:X9 + X2:X3 +
## X2:X4 + X2:X5 + X2:X8 + X2:X9 + X3:X8 + X3:X9 + X4:X8 + X4:X9 +
## X5:X8 + X5:X9 + X6:X8 + X6:X9 + X7:X8 + X7:X9 + X8:X9
##
##
## Step: AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 +
## X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X7 + X1:X8 + X1:X9 + X2:X3 +
## X2:X4 + X2:X8 + X2:X9 + X3:X8 + X3:X9 + X4:X8 + X4:X9 + X5:X8 +
## X5:X9 + X6:X8 + X6:X9 + X7:X8 + X7:X9 + X8:X9
##
##
## Step: AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 +
## X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X7 + X1:X8 + X1:X9 + X2:X3 +
## X2:X8 + X2:X9 + X3:X8 + X3:X9 + X4:X8 + X4:X9 + X5:X8 + X5:X9 +
## X6:X8 + X6:X9 + X7:X8 + X7:X9 + X8:X9
##
##
## Step: AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 +
## X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X7 + X1:X8 + X1:X9 + X2:X8 +
## X2:X9 + X3:X8 + X3:X9 + X4:X8 + X4:X9 + X5:X8 + X5:X9 + X6:X8 +
## X6:X9 + X7:X8 + X7:X9 + X8:X9
##
##
## Step: AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 +
## X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X8 + X1:X9 + X2:X8 + X2:X9 +
## X3:X8 + X3:X9 + X4:X8 + X4:X9 + X5:X8 + X5:X9 + X6:X8 + X6:X9 +
## X7:X8 + X7:X9 + X8:X9
##
##
## Step: AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 +
## X1:X3 + X1:X4 + X1:X5 + X1:X8 + X1:X9 + X2:X8 + X2:X9 + X3:X8 +
## X3:X9 + X4:X8 + X4:X9 + X5:X8 + X5:X9 + X6:X8 + X6:X9 + X7:X8 +
## X7:X9 + X8:X9
##
##
## Step: AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 +
## X1:X3 + X1:X4 + X1:X8 + X1:X9 + X2:X8 + X2:X9 + X3:X8 + X3:X9 +
## X4:X8 + X4:X9 + X5:X8 + X5:X9 + X6:X8 + X6:X9 + X7:X8 + X7:X9 +
## X8:X9
##
##
## Step: AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 +
## X1:X3 + X1:X8 + X1:X9 + X2:X8 + X2:X9 + X3:X8 + X3:X9 + X4:X8 +
## X4:X9 + X5:X8 + X5:X9 + X6:X8 + X6:X9 + X7:X8 + X7:X9 + X8:X9
##
## Df Sum of Sq RSS AIC
## - X1:X8 1 0.00125 0.27702 -88.546
## - X3:X9 1 0.00166 0.27743 -88.501
## - X1:X9 1 0.00171 0.27748 -88.496
## - X4:X9 1 0.00224 0.27801 -88.439
## - X5:X8 1 0.00375 0.27952 -88.276
## - X3:X8 1 0.01365 0.28942 -87.232
## - X5:X9 1 0.01394 0.28971 -87.202
## <none> 0.27577 -86.682
## - X4:X8 1 0.01926 0.29503 -86.656
## - X8:X9 1 0.02380 0.29957 -86.198
## - X1:X2 1 0.02492 0.30069 -86.086
## - X2:X8 1 0.02521 0.30098 -86.057
## - X6:X8 1 0.02975 0.30552 -85.608
## - X6:X9 1 0.03024 0.30601 -85.560
## - X2:X9 1 0.03404 0.30981 -85.190
## - X7:X8 1 0.04050 0.31627 -84.570
## - X7:X9 1 0.08581 0.36158 -80.554
## - X1:X3 1 1.65959 1.93536 -30.227
##
## Step: AIC=-88.55
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 +
## X1:X3 + X1:X9 + X2:X8 + X2:X9 + X3:X8 + X3:X9 + X4:X8 + X4:X9 +
## X5:X8 + X5:X9 + X6:X8 + X6:X9 + X7:X8 + X7:X9 + X8:X9
##
## Df Sum of Sq RSS AIC
## - X4:X9 1 0.00103 0.27806 -90.434
## - X5:X8 1 0.00630 0.28332 -89.871
## - X3:X9 1 0.01467 0.29170 -88.997
## <none> 0.27702 -88.546
## - X5:X9 1 0.01990 0.29692 -88.465
## - X2:X8 1 0.02658 0.30360 -87.797
## - X1:X2 1 0.02706 0.30408 -87.750
## - X3:X8 1 0.02955 0.30658 -87.504
## - X6:X9 1 0.03164 0.30866 -87.301
## - X6:X8 1 0.03412 0.31114 -87.061
## - X4:X8 1 0.03459 0.31162 -87.015
## - X2:X9 1 0.03623 0.31325 -86.859
## - X8:X9 1 0.03768 0.31470 -86.720
## - X7:X8 1 0.04267 0.31969 -86.248
## - X1:X9 1 0.08036 0.35738 -82.904
## - X7:X9 1 0.08949 0.36652 -82.147
## - X1:X3 1 1.67908 1.95611 -31.907
##
## Step: AIC=-90.43
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 +
## X1:X3 + X1:X9 + X2:X8 + X2:X9 + X3:X8 + X3:X9 + X4:X8 + X5:X8 +
## X5:X9 + X6:X8 + X6:X9 + X7:X8 + X7:X9 + X8:X9
##
## Df Sum of Sq RSS AIC
## - X5:X8 1 0.01021 0.28826 -91.352
## <none> 0.27806 -90.434
## - X3:X9 1 0.01962 0.29768 -90.388
## - X1:X2 1 0.02775 0.30580 -89.580
## - X3:X8 1 0.02855 0.30661 -89.502
## - X5:X9 1 0.02886 0.30692 -89.471
## - X6:X9 1 0.03330 0.31136 -89.040
## - X8:X9 1 0.03752 0.31557 -88.637
## - X6:X8 1 0.04082 0.31888 -88.324
## - X7:X8 1 0.05026 0.32832 -87.449
## - X2:X8 1 0.07559 0.35365 -85.220
## - X1:X9 1 0.08639 0.36445 -84.317
## - X2:X9 1 0.09477 0.37282 -83.635
## - X7:X9 1 0.09547 0.37353 -83.579
## - X4:X8 1 0.11185 0.38991 -82.291
## - X1:X3 1 1.76157 2.03963 -32.653
##
## Step: AIC=-91.35
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 +
## X1:X3 + X1:X9 + X2:X8 + X2:X9 + X3:X8 + X3:X9 + X4:X8 + X5:X9 +
## X6:X8 + X6:X9 + X7:X8 + X7:X9 + X8:X9
##
## Df Sum of Sq RSS AIC
## - X3:X9 1 0.01387 0.30213 -91.943
## <none> 0.28826 -91.352
## - X1:X2 1 0.02606 0.31432 -90.756
## - X6:X9 1 0.02892 0.31718 -90.484
## - X5:X9 1 0.03517 0.32343 -89.899
## - X6:X8 1 0.03526 0.32352 -89.891
## - X8:X9 1 0.03647 0.32474 -89.778
## - X7:X8 1 0.05418 0.34244 -88.186
## - X3:X8 1 0.06678 0.35505 -87.101
## - X1:X9 1 0.08233 0.37059 -85.815
## - X2:X8 1 0.09026 0.37852 -85.180
## - X4:X8 1 0.11594 0.40420 -83.211
## - X2:X9 1 0.12196 0.41023 -82.767
## - X7:X9 1 0.19579 0.48405 -77.803
## - X1:X3 1 1.79585 2.08412 -34.006
##
## Step: AIC=-91.94
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 +
## X1:X3 + X1:X9 + X2:X8 + X2:X9 + X3:X8 + X4:X8 + X5:X9 + X6:X8 +
## X6:X9 + X7:X8 + X7:X9 + X8:X9
##
## Df Sum of Sq RSS AIC
## - X6:X9 1 0.01568 0.31781 -92.425
## <none> 0.30213 -91.943
## - X6:X8 1 0.02200 0.32413 -91.834
## - X5:X9 1 0.02250 0.32464 -91.788
## - X1:X2 1 0.03453 0.33666 -90.696
## - X7:X8 1 0.04139 0.34352 -90.092
## - X8:X9 1 0.05081 0.35295 -89.279
## - X1:X9 1 0.07027 0.37241 -87.669
## - X2:X8 1 0.07640 0.37853 -87.180
## - X3:X8 1 0.09504 0.39717 -85.737
## - X4:X8 1 0.10557 0.40770 -84.952
## - X2:X9 1 0.10898 0.41111 -84.703
## - X7:X9 1 0.20828 0.51041 -78.212
## - X1:X3 1 1.80109 2.10322 -35.732
##
## Step: AIC=-92.43
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 +
## X1:X3 + X1:X9 + X2:X8 + X2:X9 + X3:X8 + X4:X8 + X5:X9 + X6:X8 +
## X7:X8 + X7:X9 + X8:X9
##
## Df Sum of Sq RSS AIC
## - X6:X8 1 0.00642 0.32423 -93.825
## - X1:X2 1 0.01993 0.33773 -92.601
## <none> 0.31781 -92.425
## - X5:X9 1 0.02449 0.34230 -92.198
## - X7:X8 1 0.02573 0.34354 -92.090
## - X8:X9 1 0.03582 0.35363 -91.221
## - X1:X9 1 0.06065 0.37846 -89.186
## - X2:X8 1 0.06122 0.37903 -89.140
## - X3:X8 1 0.08481 0.40262 -87.329
## - X4:X8 1 0.09252 0.41033 -86.760
## - X2:X9 1 0.09419 0.41200 -86.638
## - X7:X9 1 0.23406 0.55187 -77.869
## - X1:X3 1 1.89418 2.21199 -36.219
##
## Step: AIC=-93.83
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 +
## X1:X3 + X1:X9 + X2:X8 + X2:X9 + X3:X8 + X4:X8 + X5:X9 + X7:X8 +
## X7:X9 + X8:X9
##
## Df Sum of Sq RSS AIC
## - X7:X8 1 0.02050 0.34473 -93.986
## - X1:X2 1 0.02195 0.34617 -93.860
## <none> 0.32423 -93.825
## - X6 1 0.02936 0.35359 -93.225
## - X5:X9 1 0.03666 0.36089 -92.611
## - X8:X9 1 0.05990 0.38412 -90.740
## - X2:X8 1 0.10202 0.42624 -87.618
## - X2:X9 1 0.16870 0.49293 -83.258
## - X7:X9 1 0.23395 0.55817 -79.529
## - X1:X9 1 0.25322 0.57745 -78.510
## - X3:X8 1 0.26381 0.58803 -77.965
## - X4:X8 1 0.40165 0.72587 -71.647
## - X1:X3 1 1.90127 2.22550 -38.036
##
## Step: AIC=-93.99
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 +
## X1:X3 + X1:X9 + X2:X8 + X2:X9 + X3:X8 + X4:X8 + X5:X9 + X7:X9 +
## X8:X9
##
## Df Sum of Sq RSS AIC
## - X1:X2 1 0.01744 0.36216 -94.506
## <none> 0.34473 -93.986
## - X6 1 0.02380 0.36853 -93.983
## - X8:X9 1 0.04553 0.39026 -92.264
## - X5:X9 1 0.04913 0.39386 -91.988
## - X2:X8 1 0.08418 0.42891 -89.432
## - X2:X9 1 0.15080 0.49553 -85.100
## - X1:X9 1 0.27457 0.61930 -78.411
## - X3:X8 1 0.30355 0.64828 -77.039
## - X7:X9 1 0.32337 0.66809 -76.136
## - X4:X8 1 0.42623 0.77096 -71.840
## - X1:X3 1 1.88785 2.23258 -39.941
##
## Step: AIC=-94.51
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X3 +
## X1:X9 + X2:X8 + X2:X9 + X3:X8 + X4:X8 + X5:X9 + X7:X9 + X8:X9
##
## Df Sum of Sq RSS AIC
## - X6 1 0.02085 0.38302 -94.826
## <none> 0.36216 -94.506
## - X8:X9 1 0.03703 0.39920 -93.585
## - X5:X9 1 0.03720 0.39936 -93.573
## - X2:X8 1 0.06882 0.43098 -91.286
## - X2:X9 1 0.13369 0.49585 -87.080
## - X1:X9 1 0.26000 0.62217 -80.272
## - X3:X8 1 0.29045 0.65261 -78.839
## - X7:X9 1 0.30630 0.66847 -78.119
## - X4:X8 1 0.40893 0.77109 -73.834
## - X1:X3 1 1.87267 2.23483 -41.911
##
## Step: AIC=-94.83
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X7 + X8 + X9 + X1:X3 + X1:X9 +
## X2:X8 + X2:X9 + X3:X8 + X4:X8 + X5:X9 + X7:X9 + X8:X9
##
## Df Sum of Sq RSS AIC
## - X5:X9 1 0.02376 0.40678 -95.021
## - X8:X9 1 0.02579 0.40880 -94.871
## <none> 0.38302 -94.826
## - X2:X8 1 0.04986 0.43287 -93.155
## - X2:X9 1 0.11290 0.49591 -89.077
## - X1:X9 1 0.23967 0.62268 -82.247
## - X3:X8 1 0.28141 0.66443 -80.301
## - X7:X9 1 0.28680 0.66981 -80.059
## - X4:X8 1 0.39670 0.77972 -75.501
## - X1:X3 1 2.28557 2.66859 -38.589
##
## Step: AIC=-95.02
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X7 + X8 + X9 + X1:X3 + X1:X9 +
## X2:X8 + X2:X9 + X3:X8 + X4:X8 + X7:X9 + X8:X9
##
## Df Sum of Sq RSS AIC
## - X8:X9 1 0.02190 0.42867 -95.448
## <none> 0.40678 -95.021
## - X2:X8 1 0.02957 0.43635 -94.916
## - X2:X9 1 0.08976 0.49654 -91.039
## - X5 1 0.18602 0.59280 -85.723
## - X7:X9 1 0.26382 0.67060 -82.024
## - X1:X9 1 0.32901 0.73579 -79.240
## - X3:X8 1 0.35450 0.76128 -78.219
## - X4:X8 1 0.43472 0.84149 -75.213
## - X1:X3 1 2.29575 2.70253 -40.210
##
## Step: AIC=-95.45
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X7 + X8 + X9 + X1:X3 + X1:X9 +
## X2:X8 + X2:X9 + X3:X8 + X4:X8 + X7:X9
##
## Df Sum of Sq RSS AIC
## - X2:X8 1 0.01037 0.43904 -96.731
## <none> 0.42867 -95.448
## - X2:X9 1 0.06896 0.49763 -92.973
## - X5 1 0.17518 0.60385 -87.169
## - X7:X9 1 0.24199 0.67066 -84.021
## - X1:X9 1 0.30722 0.73589 -81.236
## - X3:X8 1 0.34794 0.77661 -79.620
## - X4:X8 1 0.47278 0.90146 -75.148
## - X1:X3 1 2.41556 2.84423 -40.677
##
## Step: AIC=-96.73
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X7 + X8 + X9 + X1:X3 + X1:X9 +
## X2:X9 + X3:X8 + X4:X8 + X7:X9
##
## Df Sum of Sq RSS AIC
## <none> 0.43904 -96.731
## - X5 1 0.17280 0.61185 -88.774
## - X7:X9 1 0.24527 0.68431 -85.416
## - X2:X9 1 0.30033 0.73937 -83.095
## - X1:X9 1 0.31184 0.75088 -82.631
## - X3:X8 1 0.40524 0.84428 -79.114
## - X4:X8 1 0.66804 1.10708 -70.984
## - X1:X3 1 2.49992 2.93897 -41.694# Interaction regression for remove X1-9
model_wf_rm1_log_inter <- lm(log(y) ~ (X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9)^2, data=table_wf)
model_wf_rm1_aic_log_inter <- stepAIC(model_wf_rm1_log_inter)
model_wf_rm2_log_inter <- lm(log(y) ~ (X1 + X3 + X4 + X5 + X6 + X7 + X8 + X9)^2, data=table_wf)
model_wf_rm2_aic_log_inter <- stepAIC(model_wf_rm2_log_inter)
model_wf_rm3_log_inter <- lm(log(y) ~ (X1 + X2 + X4 + X5 + X6 + X7+ X8 + X9)^2, data=table_wf)
model_wf_rm3_aic_log_inter <- stepAIC(model_wf_rm3_log_inter)
model_wf_rm5_log_inter <- lm(log(y) ~ (X1 + X2 + X3 + X4 + X6 + X7 + X8 + X9)^2, data=table_wf)
model_wf_rm5_aic_log_inter <- stepAIC(model_wf_rm5_log_inter)
model_wf_rm4_log_inter <- lm(log(y) ~ (X1 + X2 + X3 + X5 + X6 + X7 + X8 + X9)^2, data=table_wf)
model_wf_rm4_aic_log_inter <- stepAIC(model_wf_rm4_log_inter)
model_wf_rm6_log_inter <- lm(log(y) ~ (X2 + X3 + X1 + X5 + X4 + X7 + X8 + X9)^2, data=table_wf)
model_wf_rm6_aic_log_inter <- stepAIC(model_wf_rm6_log_inter)
model_wf_rm7_log_inter <- lm(log(y) ~ (X1 + X2 + X3 + X4 + X5 + X6 + X8 + X9)^2, data=table_wf)
model_wf_rm7_aic_log_inter <- stepAIC(model_wf_rm7_log_inter)
model_wf_rm8_log_inter <- lm(log(y) ~ (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X9)^2, data=table_wf)
model_wf_rm8_aic_log_inter <- stepAIC(model_wf_rm8_log_inter)
model_wf_rm9_log_inter <- lm(log(y) ~ (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8)^2, data=table_wf)
model_wf_rm9_aic_log_inter <- stepAIC(model_wf_rm9_log_inter)
# Interaction regression for 136789
model_wf_136789_log_inter <- lm(log(y) ~ (X3 + X1 + X6 + X7 + X8 + X9)^2, data=table_wf)
model_wf_136789_aic_log_inter <- stepAIC(model_wf_136789_log_inter)
# Interaction regression for 436789
model_wf_436789_log_inter <- lm(log(y) ~ (X3 + X4 + X6 + X7 + X8 + X9)^2, data=table_wf)
model_wf_436789_aic_log_inter <- stepAIC(model_wf_436789_log_inter)
# Interaction regression for 437
model_wf_437_log_inter <- lm(log(y) ~ (X3 + X4 + X7 )^2, data=table_wf)
model_wf_437_aic_log_inter <- stepAIC(model_wf_437_log_inter)
# Interaction regression for 489
model_wf_489_log_inter <- lm(log(y) ~ (X4 + X8 + X9 )^2, data=table_wf)
model_wf_489_aic_log_inter <- stepAIC(model_wf_489_log_inter)
# Interaction regression by groups
model_wf_3g_log_inter <- lm(log(y) ~ (X1 + X3 + X4 )^2 + (X2+ X5+X6 + X7)^2 + (X8 + X9)^2, data=table_wf)
model_wf_3g_aic_log_inter <- stepAIC(model_wf_3g_log_inter)
# Interaction regression by groups1
model_wf_3g1_log_inter <- lm(log(y) ~ (X1 + X2 + X5 )^2 + (X3 +X6 + X7)^2 + (X4 +X8 + X9)^2, data=table_wf)
model_wf_3g1_aic_log_inter <- stepAIC(model_wf_3g1_log_inter)
# Comparison
huxreg(model_wf_rm1_aic_log_inter, model_wf_rm2_aic_log_inter, model_wf_rm3_aic_log_inter, model_wf_rm4_aic_log_inter, model_wf_rm5_aic_log_inter, model_wf_rm6_aic_log_inter, model_wf_rm7_aic_log_inter, model_wf_rm8_aic_log_inter, model_wf_rm9_aic_log_inter, model_wf_aic_log_inter)
huxreg(model_wf_136789_aic_log_inter,model_wf_436789_aic_log_inter, model_wf_437_log_inter, model_wf_489_log_inter, model_wf_3g_aic_log_inter, model_wf_3g1_aic_log_inter)- All log models
# build all log model
model_wf_all_log <- lm(log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + log(X7) + log(X8) + log(X9), data=table_wf)
ols_vif_tol(model_wf_all_log)| Variables | Tolerance | VIF |
| log(X1) | 0.00608 | 164 |
| log(X2) | 0.0865 | 11.6 |
| log(X3) | 0.0816 | 12.3 |
| log(X4) | 0.00767 | 130 |
| log(X5) | 0.0885 | 11.3 |
| log(X6) | 0.421 | 2.37 |
| log(X7) | 0.108 | 9.29 |
| log(X8) | 0.193 | 5.18 |
| log(X9) | 0.187 | 5.35 |
model_wf_aic_all_log <- stepAIC(model_wf_all_log)## Start: AIC=-84.46
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) +
## log(X7) + log(X8) + log(X9)
##
## Df Sum of Sq RSS AIC
## - log(X7) 1 0.0027 0.9251 -86.371
## - log(X2) 1 0.0310 0.9534 -85.468
## - log(X4) 1 0.0370 0.9595 -85.277
## - log(X3) 1 0.0525 0.9750 -84.797
## - log(X5) 1 0.0574 0.9798 -84.647
## <none> 0.9224 -84.459
## - log(X6) 1 0.2329 1.1553 -79.705
## - log(X1) 1 0.2712 1.1936 -78.727
## - log(X9) 1 3.4818 4.4043 -39.559
## - log(X8) 1 3.6487 4.5711 -38.443
##
## Step: AIC=-86.37
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) +
## log(X8) + log(X9)
##
## Df Sum of Sq RSS AIC
## - log(X4) 1 0.0370 0.9621 -87.193
## - log(X2) 1 0.0559 0.9810 -86.611
## <none> 0.9251 -86.371
## - log(X3) 1 0.0777 1.0028 -85.953
## - log(X5) 1 0.0983 1.0234 -85.341
## - log(X6) 1 0.3174 1.2425 -79.522
## - log(X1) 1 0.3899 1.3150 -77.820
## - log(X9) 1 3.4793 4.4044 -41.558
## - log(X8) 1 3.6745 4.5996 -40.257
##
## Step: AIC=-87.19
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X5) + log(X6) + log(X8) +
## log(X9)
##
## Df Sum of Sq RSS AIC
## - log(X2) 1 0.0369 0.9990 -88.065
## <none> 0.9621 -87.193
## - log(X5) 1 0.1419 1.1040 -85.067
## - log(X6) 1 0.2985 1.2607 -81.087
## - log(X3) 1 0.5087 1.4709 -76.460
## - log(X9) 1 3.4515 4.4137 -43.495
## - log(X8) 1 3.6420 4.6042 -42.227
## - log(X1) 1 3.8134 4.7755 -41.131
##
## Step: AIC=-88.07
## log(y) ~ log(X1) + log(X3) + log(X5) + log(X6) + log(X8) + log(X9)
##
## Df Sum of Sq RSS AIC
## <none> 0.9990 -88.065
## - log(X5) 1 0.1087 1.1077 -86.967
## - log(X6) 1 0.3805 1.3795 -80.384
## - log(X3) 1 0.8252 1.8242 -72.001
## - log(X9) 1 3.4549 4.4539 -45.222
## - log(X8) 1 3.7305 4.7295 -43.421
## - log(X1) 1 17.5601 18.5592 -2.407ols_vif_tol(model_wf_aic_all_log)| Variables | Tolerance | VIF |
| log(X1) | 0.263 | 3.8 |
| log(X3) | 0.603 | 1.66 |
| log(X5) | 0.22 | 4.55 |
| log(X6) | 0.71 | 1.41 |
| log(X8) | 0.201 | 4.99 |
| log(X9) | 0.191 | 5.22 |
# Interaction regression for all log model
model_wf_all_log_inter <- lm(log(y) ~ (log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + log(X7) + log(X8) + log(X9))^2, data=table_wf)
model_wf_aic_all_log_inter <- stepAIC(model_wf_all_log_inter)## Start: AIC=-117.75
## log(y) ~ (log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) +
## log(X7) + log(X8) + log(X9))^2
##
##
## Step: AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) +
## log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) +
## log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X7) +
## log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X3) + log(X2):log(X4) +
## log(X2):log(X5) + log(X2):log(X6) + log(X2):log(X7) + log(X2):log(X8) +
## log(X2):log(X9) + log(X3):log(X4) + log(X3):log(X5) + log(X3):log(X6) +
## log(X3):log(X7) + log(X3):log(X8) + log(X3):log(X9) + log(X4):log(X5) +
## log(X4):log(X6) + log(X4):log(X7) + log(X4):log(X8) + log(X4):log(X9) +
## log(X5):log(X6) + log(X5):log(X7) + log(X5):log(X8) + log(X5):log(X9) +
## log(X6):log(X8) + log(X6):log(X9) + log(X7):log(X8) + log(X7):log(X9) +
## log(X8):log(X9)
##
##
## Step: AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) +
## log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) +
## log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X7) +
## log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X3) + log(X2):log(X4) +
## log(X2):log(X5) + log(X2):log(X6) + log(X2):log(X7) + log(X2):log(X8) +
## log(X2):log(X9) + log(X3):log(X4) + log(X3):log(X5) + log(X3):log(X6) +
## log(X3):log(X7) + log(X3):log(X8) + log(X3):log(X9) + log(X4):log(X5) +
## log(X4):log(X6) + log(X4):log(X7) + log(X4):log(X8) + log(X4):log(X9) +
## log(X5):log(X6) + log(X5):log(X8) + log(X5):log(X9) + log(X6):log(X8) +
## log(X6):log(X9) + log(X7):log(X8) + log(X7):log(X9) + log(X8):log(X9)
##
##
## Step: AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) +
## log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) +
## log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X7) +
## log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X3) + log(X2):log(X4) +
## log(X2):log(X5) + log(X2):log(X6) + log(X2):log(X7) + log(X2):log(X8) +
## log(X2):log(X9) + log(X3):log(X4) + log(X3):log(X5) + log(X3):log(X6) +
## log(X3):log(X7) + log(X3):log(X8) + log(X3):log(X9) + log(X4):log(X5) +
## log(X4):log(X6) + log(X4):log(X7) + log(X4):log(X8) + log(X4):log(X9) +
## log(X5):log(X8) + log(X5):log(X9) + log(X6):log(X8) + log(X6):log(X9) +
## log(X7):log(X8) + log(X7):log(X9) + log(X8):log(X9)
##
##
## Step: AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) +
## log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) +
## log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X7) +
## log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X3) + log(X2):log(X4) +
## log(X2):log(X5) + log(X2):log(X6) + log(X2):log(X7) + log(X2):log(X8) +
## log(X2):log(X9) + log(X3):log(X4) + log(X3):log(X5) + log(X3):log(X6) +
## log(X3):log(X7) + log(X3):log(X8) + log(X3):log(X9) + log(X4):log(X5) +
## log(X4):log(X6) + log(X4):log(X8) + log(X4):log(X9) + log(X5):log(X8) +
## log(X5):log(X9) + log(X6):log(X8) + log(X6):log(X9) + log(X7):log(X8) +
## log(X7):log(X9) + log(X8):log(X9)
##
##
## Step: AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) +
## log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) +
## log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X7) +
## log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X3) + log(X2):log(X4) +
## log(X2):log(X5) + log(X2):log(X6) + log(X2):log(X7) + log(X2):log(X8) +
## log(X2):log(X9) + log(X3):log(X4) + log(X3):log(X5) + log(X3):log(X6) +
## log(X3):log(X7) + log(X3):log(X8) + log(X3):log(X9) + log(X4):log(X5) +
## log(X4):log(X8) + log(X4):log(X9) + log(X5):log(X8) + log(X5):log(X9) +
## log(X6):log(X8) + log(X6):log(X9) + log(X7):log(X8) + log(X7):log(X9) +
## log(X8):log(X9)
##
##
## Step: AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) +
## log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) +
## log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X7) +
## log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X3) + log(X2):log(X4) +
## log(X2):log(X5) + log(X2):log(X6) + log(X2):log(X7) + log(X2):log(X8) +
## log(X2):log(X9) + log(X3):log(X4) + log(X3):log(X5) + log(X3):log(X6) +
## log(X3):log(X7) + log(X3):log(X8) + log(X3):log(X9) + log(X4):log(X8) +
## log(X4):log(X9) + log(X5):log(X8) + log(X5):log(X9) + log(X6):log(X8) +
## log(X6):log(X9) + log(X7):log(X8) + log(X7):log(X9) + log(X8):log(X9)
##
##
## Step: AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) +
## log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) +
## log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X7) +
## log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X3) + log(X2):log(X4) +
## log(X2):log(X5) + log(X2):log(X6) + log(X2):log(X7) + log(X2):log(X8) +
## log(X2):log(X9) + log(X3):log(X4) + log(X3):log(X5) + log(X3):log(X6) +
## log(X3):log(X8) + log(X3):log(X9) + log(X4):log(X8) + log(X4):log(X9) +
## log(X5):log(X8) + log(X5):log(X9) + log(X6):log(X8) + log(X6):log(X9) +
## log(X7):log(X8) + log(X7):log(X9) + log(X8):log(X9)
##
##
## Step: AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) +
## log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) +
## log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X7) +
## log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X3) + log(X2):log(X4) +
## log(X2):log(X5) + log(X2):log(X6) + log(X2):log(X7) + log(X2):log(X8) +
## log(X2):log(X9) + log(X3):log(X4) + log(X3):log(X5) + log(X3):log(X8) +
## log(X3):log(X9) + log(X4):log(X8) + log(X4):log(X9) + log(X5):log(X8) +
## log(X5):log(X9) + log(X6):log(X8) + log(X6):log(X9) + log(X7):log(X8) +
## log(X7):log(X9) + log(X8):log(X9)
##
##
## Step: AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) +
## log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) +
## log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X7) +
## log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X3) + log(X2):log(X4) +
## log(X2):log(X5) + log(X2):log(X6) + log(X2):log(X7) + log(X2):log(X8) +
## log(X2):log(X9) + log(X3):log(X4) + log(X3):log(X8) + log(X3):log(X9) +
## log(X4):log(X8) + log(X4):log(X9) + log(X5):log(X8) + log(X5):log(X9) +
## log(X6):log(X8) + log(X6):log(X9) + log(X7):log(X8) + log(X7):log(X9) +
## log(X8):log(X9)
##
##
## Step: AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) +
## log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) +
## log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X7) +
## log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X3) + log(X2):log(X4) +
## log(X2):log(X5) + log(X2):log(X6) + log(X2):log(X7) + log(X2):log(X8) +
## log(X2):log(X9) + log(X3):log(X8) + log(X3):log(X9) + log(X4):log(X8) +
## log(X4):log(X9) + log(X5):log(X8) + log(X5):log(X9) + log(X6):log(X8) +
## log(X6):log(X9) + log(X7):log(X8) + log(X7):log(X9) + log(X8):log(X9)
##
##
## Step: AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) +
## log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) +
## log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X7) +
## log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X3) + log(X2):log(X4) +
## log(X2):log(X5) + log(X2):log(X6) + log(X2):log(X8) + log(X2):log(X9) +
## log(X3):log(X8) + log(X3):log(X9) + log(X4):log(X8) + log(X4):log(X9) +
## log(X5):log(X8) + log(X5):log(X9) + log(X6):log(X8) + log(X6):log(X9) +
## log(X7):log(X8) + log(X7):log(X9) + log(X8):log(X9)
##
##
## Step: AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) +
## log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) +
## log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X7) +
## log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X3) + log(X2):log(X4) +
## log(X2):log(X5) + log(X2):log(X8) + log(X2):log(X9) + log(X3):log(X8) +
## log(X3):log(X9) + log(X4):log(X8) + log(X4):log(X9) + log(X5):log(X8) +
## log(X5):log(X9) + log(X6):log(X8) + log(X6):log(X9) + log(X7):log(X8) +
## log(X7):log(X9) + log(X8):log(X9)
##
##
## Step: AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) +
## log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) +
## log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X7) +
## log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X3) + log(X2):log(X4) +
## log(X2):log(X8) + log(X2):log(X9) + log(X3):log(X8) + log(X3):log(X9) +
## log(X4):log(X8) + log(X4):log(X9) + log(X5):log(X8) + log(X5):log(X9) +
## log(X6):log(X8) + log(X6):log(X9) + log(X7):log(X8) + log(X7):log(X9) +
## log(X8):log(X9)
##
##
## Step: AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) +
## log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) +
## log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X7) +
## log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X3) + log(X2):log(X8) +
## log(X2):log(X9) + log(X3):log(X8) + log(X3):log(X9) + log(X4):log(X8) +
## log(X4):log(X9) + log(X5):log(X8) + log(X5):log(X9) + log(X6):log(X8) +
## log(X6):log(X9) + log(X7):log(X8) + log(X7):log(X9) + log(X8):log(X9)
##
##
## Step: AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) +
## log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) +
## log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X7) +
## log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X8) + log(X2):log(X9) +
## log(X3):log(X8) + log(X3):log(X9) + log(X4):log(X8) + log(X4):log(X9) +
## log(X5):log(X8) + log(X5):log(X9) + log(X6):log(X8) + log(X6):log(X9) +
## log(X7):log(X8) + log(X7):log(X9) + log(X8):log(X9)
##
##
## Step: AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) +
## log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) +
## log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X8) +
## log(X1):log(X9) + log(X2):log(X8) + log(X2):log(X9) + log(X3):log(X8) +
## log(X3):log(X9) + log(X4):log(X8) + log(X4):log(X9) + log(X5):log(X8) +
## log(X5):log(X9) + log(X6):log(X8) + log(X6):log(X9) + log(X7):log(X8) +
## log(X7):log(X9) + log(X8):log(X9)
##
##
## Step: AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) +
## log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) +
## log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X8) + log(X1):log(X9) +
## log(X2):log(X8) + log(X2):log(X9) + log(X3):log(X8) + log(X3):log(X9) +
## log(X4):log(X8) + log(X4):log(X9) + log(X5):log(X8) + log(X5):log(X9) +
## log(X6):log(X8) + log(X6):log(X9) + log(X7):log(X8) + log(X7):log(X9) +
## log(X8):log(X9)
##
##
## Step: AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) +
## log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) +
## log(X1):log(X4) + log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X8) +
## log(X2):log(X9) + log(X3):log(X8) + log(X3):log(X9) + log(X4):log(X8) +
## log(X4):log(X9) + log(X5):log(X8) + log(X5):log(X9) + log(X6):log(X8) +
## log(X6):log(X9) + log(X7):log(X8) + log(X7):log(X9) + log(X8):log(X9)
##
##
## Step: AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) +
## log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) +
## log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X8) + log(X2):log(X9) +
## log(X3):log(X8) + log(X3):log(X9) + log(X4):log(X8) + log(X4):log(X9) +
## log(X5):log(X8) + log(X5):log(X9) + log(X6):log(X8) + log(X6):log(X9) +
## log(X7):log(X8) + log(X7):log(X9) + log(X8):log(X9)
##
## Df Sum of Sq RSS AIC
## - log(X3):log(X8) 1 0.0000903 0.097990 -119.72
## - log(X3):log(X9) 1 0.0003856 0.098286 -119.63
## - log(X7):log(X8) 1 0.0008771 0.098777 -119.48
## - log(X2):log(X9) 1 0.0009602 0.098860 -119.46
## - log(X2):log(X8) 1 0.0012711 0.099171 -119.36
## - log(X4):log(X9) 1 0.0013307 0.099231 -119.34
## - log(X6):log(X8) 1 0.0013914 0.099291 -119.33
## - log(X7):log(X9) 1 0.0014236 0.099324 -119.32
## - log(X5):log(X8) 1 0.0014490 0.099349 -119.31
## - log(X5):log(X9) 1 0.0015560 0.099456 -119.28
## - log(X4):log(X8) 1 0.0016138 0.099514 -119.26
## - log(X6):log(X9) 1 0.0016209 0.099521 -119.26
## - log(X1):log(X9) 1 0.0025113 0.100411 -118.99
## - log(X1):log(X2) 1 0.0025483 0.100448 -118.98
## - log(X8):log(X9) 1 0.0025636 0.100464 -118.97
## - log(X1):log(X8) 1 0.0029919 0.100892 -118.85
## <none> 0.097900 -117.75
## - log(X1):log(X3) 1 0.0093915 0.107292 -117.00
##
## Step: AIC=-119.72
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) +
## log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) +
## log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X8) + log(X2):log(X9) +
## log(X3):log(X9) + log(X4):log(X8) + log(X4):log(X9) + log(X5):log(X8) +
## log(X5):log(X9) + log(X6):log(X8) + log(X6):log(X9) + log(X7):log(X8) +
## log(X7):log(X9) + log(X8):log(X9)
##
## Df Sum of Sq RSS AIC
## - log(X8):log(X9) 1 0.004379 0.10237 -120.41
## - log(X1):log(X2) 1 0.004551 0.10254 -120.36
## - log(X3):log(X9) 1 0.004640 0.10263 -120.33
## <none> 0.09799 -119.72
## - log(X7):log(X8) 1 0.006852 0.10484 -119.69
## - log(X7):log(X9) 1 0.011626 0.10962 -118.36
## - log(X6):log(X8) 1 0.012314 0.11030 -118.17
## - log(X6):log(X9) 1 0.016183 0.11417 -117.14
## - log(X1):log(X9) 1 0.022498 0.12049 -115.52
## - log(X4):log(X9) 1 0.024248 0.12224 -115.09
## - log(X1):log(X8) 1 0.025269 0.12326 -114.84
## - log(X2):log(X9) 1 0.025677 0.12367 -114.74
## - log(X1):log(X3) 1 0.027330 0.12532 -114.34
## - log(X5):log(X9) 1 0.029018 0.12701 -113.94
## - log(X4):log(X8) 1 0.030440 0.12843 -113.61
## - log(X5):log(X8) 1 0.031030 0.12902 -113.47
## - log(X2):log(X8) 1 0.034946 0.13294 -112.57
##
## Step: AIC=-120.41
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) +
## log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) +
## log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X8) + log(X2):log(X9) +
## log(X3):log(X9) + log(X4):log(X8) + log(X4):log(X9) + log(X5):log(X8) +
## log(X5):log(X9) + log(X6):log(X8) + log(X6):log(X9) + log(X7):log(X8) +
## log(X7):log(X9)
##
## Df Sum of Sq RSS AIC
## - log(X1):log(X2) 1 0.0019105 0.10428 -121.86
## - log(X7):log(X8) 1 0.0042768 0.10665 -121.18
## <none> 0.10237 -120.41
## - log(X7):log(X9) 1 0.0085205 0.11089 -120.01
## - log(X6):log(X8) 1 0.0088118 0.11118 -119.93
## - log(X6):log(X9) 1 0.0122126 0.11458 -119.03
## - log(X1):log(X9) 1 0.0181216 0.12049 -117.52
## - log(X4):log(X9) 1 0.0203055 0.12268 -116.98
## - log(X3):log(X9) 1 0.0205839 0.12295 -116.91
## - log(X1):log(X8) 1 0.0209330 0.12330 -116.83
## - log(X2):log(X9) 1 0.0213148 0.12368 -116.74
## - log(X1):log(X3) 1 0.0237155 0.12609 -116.16
## - log(X5):log(X9) 1 0.0249321 0.12730 -115.87
## - log(X4):log(X8) 1 0.0267336 0.12910 -115.45
## - log(X5):log(X8) 1 0.0270663 0.12944 -115.37
## - log(X2):log(X8) 1 0.0305761 0.13295 -114.57
##
## Step: AIC=-121.86
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) +
## log(X7) + log(X8) + log(X9) + log(X1):log(X3) + log(X1):log(X8) +
## log(X1):log(X9) + log(X2):log(X8) + log(X2):log(X9) + log(X3):log(X9) +
## log(X4):log(X8) + log(X4):log(X9) + log(X5):log(X8) + log(X5):log(X9) +
## log(X6):log(X8) + log(X6):log(X9) + log(X7):log(X8) + log(X7):log(X9)
##
## Df Sum of Sq RSS AIC
## - log(X7):log(X8) 1 0.0055209 0.10980 -122.31
## <none> 0.10428 -121.86
## - log(X7):log(X9) 1 0.0100914 0.11437 -121.08
## - log(X6):log(X8) 1 0.0111603 0.11544 -120.81
## - log(X6):log(X9) 1 0.0144674 0.11875 -119.96
## - log(X2):log(X9) 1 0.0220327 0.12631 -118.11
## - log(X1):log(X9) 1 0.0224953 0.12677 -118.00
## - log(X3):log(X9) 1 0.0225020 0.12678 -118.00
## - log(X4):log(X9) 1 0.0227039 0.12698 -117.95
## - log(X1):log(X3) 1 0.0232574 0.12754 -117.82
## - log(X5):log(X9) 1 0.0244629 0.12874 -117.53
## - log(X5):log(X8) 1 0.0257774 0.13006 -117.23
## - log(X1):log(X8) 1 0.0262811 0.13056 -117.11
## - log(X4):log(X8) 1 0.0296051 0.13389 -116.36
## - log(X2):log(X8) 1 0.0312323 0.13551 -116.00
##
## Step: AIC=-122.31
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) +
## log(X7) + log(X8) + log(X9) + log(X1):log(X3) + log(X1):log(X8) +
## log(X1):log(X9) + log(X2):log(X8) + log(X2):log(X9) + log(X3):log(X9) +
## log(X4):log(X8) + log(X4):log(X9) + log(X5):log(X8) + log(X5):log(X9) +
## log(X6):log(X8) + log(X6):log(X9) + log(X7):log(X9)
##
## Df Sum of Sq RSS AIC
## <none> 0.10980 -122.309
## - log(X3):log(X9) 1 0.020900 0.13070 -119.081
## - log(X1):log(X3) 1 0.021290 0.13109 -118.992
## - log(X7):log(X9) 1 0.023868 0.13367 -118.408
## - log(X6):log(X8) 1 0.030444 0.14024 -116.967
## - log(X5):log(X8) 1 0.032198 0.14200 -116.594
## - log(X1):log(X9) 1 0.034511 0.14431 -116.109
## - log(X5):log(X9) 1 0.036577 0.14638 -115.683
## - log(X6):log(X9) 1 0.057094 0.16689 -111.748
## - log(X1):log(X8) 1 0.058955 0.16876 -111.415
## - log(X2):log(X9) 1 0.067720 0.17752 -109.896
## - log(X4):log(X9) 1 0.086014 0.19581 -106.953
## - log(X2):log(X8) 1 0.117909 0.22771 -102.426
## - log(X4):log(X8) 1 0.199440 0.30924 -93.245huxreg(model_wf_aic_log, model_wf_aic_all_log, model_wf_aic_log_inter, model_wf_aic_all_log_inter)| (1) | (2) | (3) | (4) | |
| (Intercept) | 2.692 *** | 0.571 | -0.981 | -16.027 |
| (0.445) | (3.360) | (1.520) | (13.947) | |
| X3 | 0.184 *** | 0.493 *** | ||
| (0.032) | (0.066) | |||
| X4 | 0.109 *** | 0.168 ** | ||
| (0.026) | (0.054) | |||
| X6 | -0.368 * | |||
| (0.146) | ||||
| X7 | 4.085 ** | 2.515 | ||
| (1.213) | (1.258) | |||
| X8 | 0.612 *** | 0.586 *** | ||
| (0.133) | (0.087) | |||
| X9 | -0.448 *** | -0.248 * | ||
| (0.108) | (0.105) | |||
| log(X1) | 0.726 *** | 1.158 | ||
| (0.036) | (0.697) | |||
| log(X3) | 0.419 *** | 1.666 | ||
| (0.096) | (1.763) | |||
| log(X5) | 1.259 | 5.028 | ||
| (0.796) | (3.473) | |||
| log(X6) | -0.267 ** | -0.158 | ||
| (0.090) | (0.300) | |||
| log(X8) | 1.623 *** | 44.919 | ||
| (0.175) | (24.175) | |||
| log(X9) | -1.375 *** | -37.066 | ||
| (0.154) | (19.170) | |||
| X1 | 2.712 *** | |||
| (0.330) | ||||
| X2 | -24.452 *** | |||
| (5.631) | ||||
| X5 | 0.039 * | |||
| (0.016) | ||||
| X1:X3 | -0.416 *** | |||
| (0.045) | ||||
| X1:X9 | -0.118 ** | |||
| (0.036) | ||||
| X2:X9 | 4.643 ** | |||
| (1.449) | ||||
| X3:X8 | -0.051 ** | |||
| (0.014) | ||||
| X4:X8 | 0.071 *** | |||
| (0.015) | ||||
| X7:X9 | -0.995 * | |||
| (0.344) | ||||
| log(X2) | -0.733 | |||
| (0.338) | ||||
| log(X4) | -2.394 | |||
| (2.963) | ||||
| log(X7) | -0.408 | |||
| (0.518) | ||||
| log(X1):log(X3) | 0.823 | |||
| (0.707) | ||||
| log(X1):log(X8) | 1.242 | |||
| (0.641) | ||||
| log(X1):log(X9) | -1.000 | |||
| (0.674) | ||||
| log(X2):log(X8) | 1.089 * | |||
| (0.397) | ||||
| log(X2):log(X9) | -0.710 | |||
| (0.341) | ||||
| log(X3):log(X9) | 0.411 | |||
| (0.356) | ||||
| log(X4):log(X8) | -3.029 ** | |||
| (0.849) | ||||
| log(X4):log(X9) | 2.174 | |||
| (0.929) | ||||
| log(X5):log(X8) | -7.969 | |||
| (5.562) | ||||
| log(X5):log(X9) | 6.795 | |||
| (4.450) | ||||
| log(X6):log(X8) | 0.403 | |||
| (0.289) | ||||
| log(X6):log(X9) | -0.506 | |||
| (0.265) | ||||
| log(X7):log(X9) | 0.464 | |||
| (0.376) | ||||
| N | 30 | 30 | 30 | 30 |
| R2 | 0.947 | 0.986 | 0.994 | 0.998 |
| logLik | -11.581 | 8.464 | 20.797 | 41.586 |
| AIC | 39.161 | -0.929 | -9.594 | -35.172 |
| *** p < 0.001; ** p < 0.01; * p < 0.05. | ||||
- Mixed models
# Mixed regression 1
model_wf_mix1 <- lm(log(y) ~ (X1 + X3 + X4 )^2 + log(X2+ X5+X6 + X7) + log(X8) + log(X9), data=table_wf)
model_wf_aic_mix1 <- stepAIC(model_wf_mix1)## Start: AIC=-80.71
## log(y) ~ (X1 + X3 + X4)^2 + log(X2 + X5 + X6 + X7) + log(X8) +
## log(X9)
##
##
## Step: AIC=-80.71
## log(y) ~ X1 + X3 + X4 + log(X2 + X5 + X6 + X7) + log(X8) + log(X9) +
## X1:X3 + X1:X4
##
## Df Sum of Sq RSS AIC
## - log(X2 + X5 + X6 + X7) 1 0.0122 1.1294 -82.385
## <none> 1.1172 -80.711
## - X1:X4 1 0.2938 1.4110 -75.706
## - X1:X3 1 1.8665 2.9837 -53.241
## - log(X9) 1 3.2544 4.3716 -41.782
## - log(X8) 1 3.4346 4.5518 -40.570
##
## Step: AIC=-82.38
## log(y) ~ X1 + X3 + X4 + log(X8) + log(X9) + X1:X3 + X1:X4
##
## Df Sum of Sq RSS AIC
## <none> 1.1294 -82.385
## - X1:X4 1 0.3135 1.4429 -77.036
## - X1:X3 1 2.3929 3.5224 -50.262
## - log(X9) 1 3.9030 5.0324 -39.559
## - log(X8) 1 4.3068 5.4362 -37.243# Mixed regression 2
model_wf_mix2 <- lm(log(y) ~ (X1 + X3 + X4 )^2 + log(X2)+log(X5)+log(X6)+log(X7) + log(X8) + log(X9), data=table_wf)
model_wf_aic_mix2 <- stepAIC(model_wf_mix2)## Start: AIC=-81.81
## log(y) ~ (X1 + X3 + X4)^2 + log(X2) + log(X5) + log(X6) + log(X7) +
## log(X8) + log(X9)
##
##
## Step: AIC=-81.81
## log(y) ~ X1 + X3 + X4 + log(X2) + log(X5) + log(X6) + log(X7) +
## log(X8) + log(X9) + X1:X3 + X1:X4
##
## Df Sum of Sq RSS AIC
## - log(X6) 1 0.0002 0.8820 -83.802
## - log(X5) 1 0.0082 0.8900 -83.531
## - log(X2) 1 0.0112 0.8930 -83.431
## - log(X7) 1 0.0187 0.9005 -83.181
## - X1:X4 1 0.0379 0.9197 -82.547
## <none> 0.8818 -81.810
## - X1:X3 1 0.3016 1.1834 -74.984
## - log(X9) 1 3.4852 4.3670 -35.813
## - log(X8) 1 3.6405 4.5223 -34.765
##
## Step: AIC=-83.8
## log(y) ~ X1 + X3 + X4 + log(X2) + log(X5) + log(X7) + log(X8) +
## log(X9) + X1:X3 + X1:X4
##
## Df Sum of Sq RSS AIC
## - log(X5) 1 0.0305 0.9125 -84.783
## - log(X2) 1 0.0553 0.9374 -83.977
## <none> 0.8820 -83.802
## - log(X7) 1 0.1162 0.9982 -82.091
## - X1:X4 1 0.1973 1.0793 -79.746
## - X1:X3 1 1.9085 2.7905 -51.249
## - log(X9) 1 3.4883 4.3704 -37.791
## - log(X8) 1 3.6480 4.5300 -36.714
##
## Step: AIC=-84.78
## log(y) ~ X1 + X3 + X4 + log(X2) + log(X7) + log(X8) + log(X9) +
## X1:X3 + X1:X4
##
## Df Sum of Sq RSS AIC
## - log(X2) 1 0.0277 0.9402 -85.886
## <none> 0.9125 -84.783
## - log(X7) 1 0.1121 1.0246 -83.308
## - X1:X4 1 0.1731 1.0856 -81.571
## - X1:X3 1 1.8806 2.7930 -53.222
## - log(X9) 1 3.6887 4.6012 -38.246
## - log(X8) 1 4.1243 5.0368 -35.533
##
## Step: AIC=-85.89
## log(y) ~ X1 + X3 + X4 + log(X7) + log(X8) + log(X9) + X1:X3 +
## X1:X4
##
## Df Sum of Sq RSS AIC
## <none> 0.9402 -85.886
## - log(X7) 1 0.1892 1.1294 -82.385
## - X1:X4 1 0.2211 1.1613 -81.549
## - X1:X3 1 2.5494 3.4896 -48.542
## - log(X9) 1 4.0912 5.0314 -37.565
## - log(X8) 1 4.4859 5.4261 -35.300# Mixed regression 3
model_wf_mix3 <- lm(log(y) ~ (X1 + X3 + X4 )^2 + (log(X2)+log(X5)+log(X6)+log(X7))^2 + log(X8) + log(X9), data=table_wf)
model_wf_aic_mix3 <- stepAIC(model_wf_mix3)## Start: AIC=-81.81
## log(y) ~ (X1 + X3 + X4)^2 + (log(X2) + log(X5) + log(X6) + log(X7))^2 +
## log(X8) + log(X9)
##
##
## Step: AIC=-81.81
## log(y) ~ X1 + X3 + X4 + log(X2) + log(X5) + log(X6) + log(X7) +
## log(X8) + log(X9) + X1:X3 + X1:X4 + X3:X4 + log(X2):log(X5) +
## log(X2):log(X6) + log(X2):log(X7) + log(X5):log(X6) + log(X5):log(X7)
##
##
## Step: AIC=-81.81
## log(y) ~ X1 + X3 + X4 + log(X2) + log(X5) + log(X6) + log(X7) +
## log(X8) + log(X9) + X1:X3 + X1:X4 + X3:X4 + log(X2):log(X5) +
## log(X2):log(X6) + log(X2):log(X7) + log(X5):log(X6)
##
##
## Step: AIC=-81.81
## log(y) ~ X1 + X3 + X4 + log(X2) + log(X5) + log(X6) + log(X7) +
## log(X8) + log(X9) + X1:X3 + X1:X4 + X3:X4 + log(X2):log(X5) +
## log(X2):log(X6) + log(X2):log(X7)
##
##
## Step: AIC=-81.81
## log(y) ~ X1 + X3 + X4 + log(X2) + log(X5) + log(X6) + log(X7) +
## log(X8) + log(X9) + X1:X3 + X1:X4 + X3:X4 + log(X2):log(X5) +
## log(X2):log(X6)
##
##
## Step: AIC=-81.81
## log(y) ~ X1 + X3 + X4 + log(X2) + log(X5) + log(X6) + log(X7) +
## log(X8) + log(X9) + X1:X3 + X1:X4 + X3:X4 + log(X2):log(X5)
##
##
## Step: AIC=-81.81
## log(y) ~ X1 + X3 + X4 + log(X2) + log(X5) + log(X6) + log(X7) +
## log(X8) + log(X9) + X1:X3 + X1:X4 + X3:X4
##
##
## Step: AIC=-81.81
## log(y) ~ X1 + X3 + X4 + log(X2) + log(X5) + log(X6) + log(X7) +
## log(X8) + log(X9) + X1:X3 + X1:X4
##
## Df Sum of Sq RSS AIC
## - log(X6) 1 0.0002 0.8820 -83.802
## - log(X5) 1 0.0082 0.8900 -83.531
## - log(X2) 1 0.0112 0.8930 -83.431
## - log(X7) 1 0.0187 0.9005 -83.181
## - X1:X4 1 0.0379 0.9197 -82.547
## <none> 0.8818 -81.810
## - X1:X3 1 0.3016 1.1834 -74.984
## - log(X9) 1 3.4852 4.3670 -35.813
## - log(X8) 1 3.6405 4.5223 -34.765
##
## Step: AIC=-83.8
## log(y) ~ X1 + X3 + X4 + log(X2) + log(X5) + log(X7) + log(X8) +
## log(X9) + X1:X3 + X1:X4
##
## Df Sum of Sq RSS AIC
## - log(X5) 1 0.0305 0.9125 -84.783
## - log(X2) 1 0.0553 0.9374 -83.977
## <none> 0.8820 -83.802
## - log(X7) 1 0.1162 0.9982 -82.091
## - X1:X4 1 0.1973 1.0793 -79.746
## - X1:X3 1 1.9085 2.7905 -51.249
## - log(X9) 1 3.4883 4.3704 -37.791
## - log(X8) 1 3.6480 4.5300 -36.714
##
## Step: AIC=-84.78
## log(y) ~ X1 + X3 + X4 + log(X2) + log(X7) + log(X8) + log(X9) +
## X1:X3 + X1:X4
##
## Df Sum of Sq RSS AIC
## - log(X2) 1 0.0277 0.9402 -85.886
## <none> 0.9125 -84.783
## - log(X7) 1 0.1121 1.0246 -83.308
## - X1:X4 1 0.1731 1.0856 -81.571
## - X1:X3 1 1.8806 2.7930 -53.222
## - log(X9) 1 3.6887 4.6012 -38.246
## - log(X8) 1 4.1243 5.0368 -35.533
##
## Step: AIC=-85.89
## log(y) ~ X1 + X3 + X4 + log(X7) + log(X8) + log(X9) + X1:X3 +
## X1:X4
##
## Df Sum of Sq RSS AIC
## <none> 0.9402 -85.886
## - log(X7) 1 0.1892 1.1294 -82.385
## - X1:X4 1 0.2211 1.1613 -81.549
## - X1:X3 1 2.5494 3.4896 -48.542
## - log(X9) 1 4.0912 5.0314 -37.565
## - log(X8) 1 4.4859 5.4261 -35.300huxreg(model_wf_aic_mix1, model_wf_aic_mix2, model_wf_aic_mix3)| (1) | (2) | (3) | |
| (Intercept) | 2.598 *** | 2.044 *** | 2.044 *** |
| (0.228) | (0.343) | (0.343) | |
| X1 | 1.290 *** | 1.459 *** | 1.459 *** |
| (0.232) | (0.232) | (0.232) | |
| X3 | 0.296 *** | 0.275 *** | 0.275 *** |
| (0.040) | (0.039) | (0.039) | |
| X4 | 0.391 *** | 0.423 *** | 0.423 *** |
| (0.042) | (0.042) | (0.042) | |
| log(X8) | 1.575 *** | 1.628 *** | 1.628 *** |
| (0.172) | (0.163) | (0.163) | |
| log(X9) | -1.345 *** | -1.405 *** | -1.405 *** |
| (0.154) | (0.147) | (0.147) | |
| X1:X3 | -0.381 *** | -0.398 *** | -0.398 *** |
| (0.056) | (0.053) | (0.053) | |
| X1:X4 | 0.031 * | 0.026 * | 0.026 * |
| (0.012) | (0.012) | (0.012) | |
| log(X7) | -0.428 | -0.428 | |
| (0.208) | (0.208) | ||
| N | 30 | 30 | 30 |
| R2 | 0.984 | 0.987 | 0.987 |
| logLik | 6.624 | 9.375 | 9.375 |
| AIC | 4.752 | 1.250 | 1.250 |
| *** p < 0.001; ** p < 0.01; * p < 0.05. | |||
Additional Steps
- step both
# Stepwise Regression based on p values for full model#
k <- ols_step_both_p(model_wf_full_log)## Stepwise Selection Method
## ---------------------------
##
## Candidate Terms:
##
## 1. X1
## 2. X2
## 3. X3
## 4. X4
## 5. X5
## 6. X6
## 7. X7
## 8. X8
## 9. X9
##
## We are selecting variables based on p value...
##
## Variables Entered/Removed:
##
## - X4 added
## - X3 added
## - X7 added
##
## No more variables to be added/removed.
##
##
## Final Model Output
## ------------------
##
## Model Summary
## -------------------------------------------------------------
## R 0.944 RMSE 0.549
## R-Squared 0.890 Coef. Var 8.618
## Adj. R-Squared 0.878 MSE 0.301
## Pred R-Squared 0.854 MAE 0.414
## -------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## -------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## -------------------------------------------------------------------
## Regression 63.565 3 21.188 70.378 0.0000
## Residual 7.828 26 0.301
## Total 71.393 29
## -------------------------------------------------------------------
##
## Parameter Estimates
## -------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## -------------------------------------------------------------------------------------
## (Intercept) 2.872 0.547 5.254 0.000 1.748 3.995
## X4 0.122 0.033 0.559 3.730 0.001 0.055 0.189
## X3 0.168 0.040 0.435 4.165 0.000 0.085 0.251
## X7 3.106 1.537 0.309 2.021 0.054 -0.053 6.266
## ------------------------------------------------------------------------------------- k##
## Stepwise Selection Summary
## ------------------------------------------------------------------------------------
## Added/ Adj.
## Step Variable Removed R-Square R-Square C(p) AIC RMSE
## ------------------------------------------------------------------------------------
## 1 X4 addition 0.803 0.796 48.8550 68.4060 0.7087
## 2 X3 addition 0.873 0.864 24.2130 57.2082 0.5792
## 3 X7 addition 0.890 0.878 19.6670 54.8305 0.5487
## ------------------------------------------------------------------------------------# plot(k)
# Stepwise AIC Regression for full model#
k<- ols_step_both_aic(model_wf_full_log)## Stepwise Selection Method
## -------------------------
##
## Candidate Terms:
##
## 1 . X1
## 2 . X2
## 3 . X3
## 4 . X4
## 5 . X5
## 6 . X6
## 7 . X7
## 8 . X8
## 9 . X9
##
##
## Variables Entered/Removed:
##
## - X4 added
## - X3 added
## - X7 added
## - X8 added
## - X9 added
## - X6 added
##
## No more variables to be added or removed. k##
##
## Stepwise Summary
## --------------------------------------------------------------------------
## Variable Method AIC RSS Sum Sq R-Sq Adj. R-Sq
## --------------------------------------------------------------------------
## X4 addition 68.406 14.063 57.330 0.80302 0.79599
## X3 addition 57.208 9.057 62.335 0.87313 0.86373
## X7 addition 54.830 7.828 63.565 0.89036 0.87771
## X8 addition 54.522 7.248 64.144 0.89848 0.88223
## X9 addition 44.504 4.856 66.537 0.93199 0.91782
## X6 addition 39.161 3.801 67.591 0.94675 0.93286
## --------------------------------------------------------------------------# plot(k)
# Stepwise Regression based on p values for all log model #
k <- ols_step_both_p(model_wf_all_log)## Stepwise Selection Method
## ---------------------------
##
## Candidate Terms:
##
## 1. log(X1)
## 2. log(X2)
## 3. log(X3)
## 4. log(X4)
## 5. log(X5)
## 6. log(X6)
## 7. log(X7)
## 8. log(X8)
## 9. log(X9)
##
## We are selecting variables based on p value...
##
## Variables Entered/Removed:
##
## - log(X4) added
##
## No more variables to be added/removed.
##
##
## Final Model Output
## ------------------
##
## Model Summary
## -------------------------------------------------------------
## R 0.954 RMSE 0.479
## R-Squared 0.910 Coef. Var 7.526
## Adj. R-Squared 0.907 MSE 0.230
## Pred R-Squared 0.896 MAE 0.353
## -------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## --------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## --------------------------------------------------------------------
## Regression 64.964 1 64.964 282.937 0.0000
## Residual 6.429 28 0.230
## Total 71.393 29
## --------------------------------------------------------------------
##
## Parameter Estimates
## -------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## -------------------------------------------------------------------------------------
## (Intercept) 4.189 0.156 26.803 0.000 3.868 4.509
## log(X4) 1.259 0.075 0.954 16.821 0.000 1.106 1.413
## -------------------------------------------------------------------------------------k##
## Stepwise Selection Summary
## -------------------------------------------------------------------------------------
## Added/ Adj.
## Step Variable Removed R-Square R-Square C(p) AIC RMSE
## -------------------------------------------------------------------------------------
## 1 log(X4) addition 0.910 0.907 113.3920 44.9246 0.4792
## -------------------------------------------------------------------------------------#plot(k)
# Stepwise AIC Regression for all log model #
k <- ols_step_both_aic(model_wf_all_log)## Stepwise Selection Method
## -------------------------
##
## Candidate Terms:
##
## 1 . log(X1)
## 2 . log(X2)
## 3 . log(X3)
## 4 . log(X4)
## 5 . log(X5)
## 6 . log(X6)
## 7 . log(X7)
## 8 . log(X8)
## 9 . log(X9)
##
##
## Variables Entered/Removed:
##
## - log(X4) added
## - log(X8) added
## - log(X9) added
## - log(X6) added
## - log(X1) added
## - log(X3) added
##
## No more variables to be added or removed.k##
##
## Stepwise Summary
## -------------------------------------------------------------------------
## Variable Method AIC RSS Sum Sq R-Sq Adj. R-Sq
## -------------------------------------------------------------------------
## log(X4) addition 44.925 6.429 64.964 0.90995 0.90673
## log(X8) addition 44.798 5.989 65.404 0.91611 0.90990
## log(X9) addition 14.099 2.014 69.379 0.97179 0.96854
## log(X6) addition 4.579 1.372 70.021 0.98079 0.97771
## log(X1) addition 2.166 1.184 70.209 0.98342 0.97996
## log(X3) addition 0.009 1.031 70.362 0.98556 0.98180
## -------------------------------------------------------------------------# plot(k)
# Stepwise Regression based on p values for all log model #
k <- ols_step_both_p(model_wf_full_log_inter)## Stepwise Selection Method
## ---------------------------
##
## Candidate Terms:
##
## 1. X1
## 2. X2
## 3. X3
## 4. X4
## 5. X5
## 6. X6
## 7. X7
## 8. X8
## 9. X9
## 10. X1:X2
## 11. X1:X3
## 12. X1:X4
## 13. X1:X5
## 14. X1:X6
## 15. X1:X7
## 16. X1:X8
## 17. X1:X9
## 18. X2:X3
## 19. X2:X4
## 20. X2:X5
## 21. X2:X6
## 22. X2:X7
## 23. X2:X8
## 24. X2:X9
## 25. X3:X4
## 26. X3:X5
## 27. X3:X6
## 28. X3:X7
## 29. X3:X8
## 30. X3:X9
## 31. X4:X5
## 32. X4:X6
## 33. X4:X7
## 34. X4:X8
## 35. X4:X9
## 36. X5:X6
## 37. X5:X7
## 38. X5:X8
## 39. X5:X9
## 40. X6:X7
## 41. X6:X8
## 42. X6:X9
## 43. X7:X8
## 44. X7:X9
## 45. X8:X9
##
## We are selecting variables based on p value...
##
## Variables Entered/Removed:
##
## - X3:X7 added
## - X6 added
## - X8 added
## - X9 added
## - X3 added
## - X5 added
## - X1 added
## - X3 added
## - X7 added
## - X9 added
## - X2 added
## - X1 added
## - X4 added
## - X8 added
## - X3:X4 added
## - X3:X7 added
## - X3:X5 added
## - X6 added
## - X1:X8 added
## - X5 added
## - X6:X9 added
## - X2 added
## - X6:X8 added
## - X1:X4 added
## - X7 added
##
## No more variables to be added/removed.
##
##
## Final Model Output
## ------------------
##
## Model Summary
## -------------------------------------------------------------
## R 0.991 RMSE 0.237
## R-Squared 0.983 Coef. Var 3.727
## Adj. R-Squared 0.977 MSE 0.056
## Pred R-Squared 0.961 MAE 0.153
## -------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## --------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## --------------------------------------------------------------------
## Regression 70.154 7 10.022 178.035 0.0000
## Residual 1.238 22 0.056
## Total 71.393 29
## --------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) 2.602 0.256 10.163 0.000 2.071 3.134
## X4 0.411 0.036 1.881 11.319 0.000 0.336 0.486
## X4:X3 -0.031 0.005 -1.207 -6.568 0.000 -0.041 -0.022
## X3:X5 0.005 0.001 0.860 7.218 0.000 0.004 0.007
## X1:X8 0.048 0.010 0.313 4.597 0.000 0.026 0.070
## X6:X9 -0.303 0.045 -0.584 -6.769 0.000 -0.395 -0.210
## X8:X6 0.297 0.054 0.486 5.489 0.000 0.185 0.409
## X4:X1 -0.012 0.003 -0.398 -3.689 0.001 -0.019 -0.005
## ----------------------------------------------------------------------------------------k##
## Stepwise Selection Summary
## ------------------------------------------------------------------------------------
## Added/ Adj.
## Step Variable Removed R-Square R-Square C(p) AIC RMSE
## ------------------------------------------------------------------------------------
## 1 X3:X7 addition 0.847 0.842 92.7180 60.7989 0.6243
## 2 X6 addition 0.862 0.851 83.4120 59.7966 0.6047
## 3 X8 addition 0.873 0.859 76.3490 59.1522 0.5897
## 4 X9 addition 0.914 0.900 46.6570 49.4831 0.4951
## 5 X3 addition 0.915 0.897 48.3380 51.3395 0.5041
## 6 X5 addition 0.927 0.908 40.9360 48.7543 0.4770
## 7 X1 addition 0.950 0.934 24.6230 39.1116 0.4017
## 8 X3 removal 0.934 0.917 34.9220 45.4055 0.4511
## 9 X7 addition 0.953 0.938 22.6940 37.5750 0.3916
## 10 X9 removal 0.918 0.896 47.8030 52.1707 0.5050
## 11 X2 addition 0.919 0.893 49.0940 53.8354 0.5134
## 12 X1 removal 0.893 0.865 66.9630 60.0481 0.5758
## 13 X4 addition 0.915 0.887 52.3760 55.3564 0.5266
## 14 X8 removal 0.906 0.881 57.1400 56.2676 0.5407
## 15 X3:X4 addition 0.934 0.913 37.2840 47.6179 0.4629
## 16 X3:X7 removal 0.861 0.825 91.9210 67.9385 0.6568
## 17 X3:X5 addition 0.934 0.913 37.3040 47.6297 0.4630
## 18 X6 removal 0.934 0.917 35.3190 45.6383 0.4529
## 19 X1:X8 addition 0.946 0.929 27.6400 41.3686 0.4171
## 20 X5 removal 0.940 0.924 30.8160 42.8830 0.4326
## 21 X6:X9 addition 0.964 0.953 13.8780 29.3316 0.3413
## 22 X2 removal 0.964 0.955 11.8800 27.3333 0.3338
## 23 X6:X8 addition 0.979 0.972 2.2430 13.1258 0.2605
## 24 X1:X4 addition 0.984 0.977 0.7850 7.9454 0.2366
## 25 X7 removal 0.983 0.977 -0.5280 7.5158 0.2373
## ------------------------------------------------------------------------------------# plot(k)
# Stepwise AIC Regression for all log model #
k <- ols_step_both_aic(model_wf_full_log_inter)## Stepwise Selection Method
## -------------------------
##
## Candidate Terms:
##
## 1 . X1
## 2 . X2
## 3 . X3
## 4 . X4
## 5 . X5
## 6 . X6
## 7 . X7
## 8 . X8
## 9 . X9
## 10 . X1:X2
## 11 . X1:X3
## 12 . X1:X4
## 13 . X1:X5
## 14 . X1:X6
## 15 . X1:X7
## 16 . X1:X8
## 17 . X1:X9
## 18 . X2:X3
## 19 . X2:X4
## 20 . X2:X5
## 21 . X2:X6
## 22 . X2:X7
## 23 . X2:X8
## 24 . X2:X9
## 25 . X3:X4
## 26 . X3:X5
## 27 . X3:X6
## 28 . X3:X7
## 29 . X3:X8
## 30 . X3:X9
## 31 . X4:X5
## 32 . X4:X6
## 33 . X4:X7
## 34 . X4:X8
## 35 . X4:X9
## 36 . X5:X6
## 37 . X5:X7
## 38 . X5:X8
## 39 . X5:X9
## 40 . X6:X7
## 41 . X6:X8
## 42 . X6:X9
## 43 . X7:X8
## 44 . X7:X9
## 45 . X8:X9
##
##
## Variables Entered/Removed:
##
## - X3:X7 added
## - X4:X5 added
## - X3:X5 added
## - X3:X4 added
## - X3:X7 removed
## - X1:X8 added
## - X9 added
## - X8 added
## - X6:X9 added
## - X6:X7 added
## - X3:X9 added
## - X3:X8 added
##
## No more variables to be added or removed.k##
##
## Stepwise Summary
## --------------------------------------------------------------------------
## Variable Method AIC RSS Sum Sq R-Sq Adj. R-Sq
## --------------------------------------------------------------------------
## X3:X7 addition 60.799 10.913 60.480 0.84714 0.84168
## X4:X5 addition 54.665 8.321 63.071 0.88344 0.87481
## X3:X5 addition 50.611 6.801 64.592 0.90474 0.89375
## X3:X4 addition 46.641 5.574 65.819 0.92193 0.90944
## X3:X7 removal 44.948 5.631 65.762 0.92113 0.91203
## X1:X8 addition 37.518 4.112 67.280 0.94240 0.93319
## X9 addition 24.336 2.479 68.914 0.96528 0.95804
## X8 addition 6.182 1.266 70.126 0.98226 0.97764
## X6:X9 addition 3.316 1.077 70.316 0.98492 0.98012
## X6:X7 addition -4.379 0.779 70.613 0.98908 0.98493
## X3:X9 addition -4.884 0.717 70.676 0.98996 0.98544
## X3:X8 addition -5.046 0.667 70.725 0.99066 0.98574
## --------------------------------------------------------------------------# plot(k)
# Stepwise Regression based on p values for all log model #
k <- ols_step_both_p(model_wf_all_log_inter)## Stepwise Selection Method
## ---------------------------
##
## Candidate Terms:
##
## 1. log(X1)
## 2. log(X2)
## 3. log(X3)
## 4. log(X4)
## 5. log(X5)
## 6. log(X6)
## 7. log(X7)
## 8. log(X8)
## 9. log(X9)
## 10. log(X1):log(X2)
## 11. log(X1):log(X3)
## 12. log(X1):log(X4)
## 13. log(X1):log(X5)
## 14. log(X1):log(X6)
## 15. log(X1):log(X7)
## 16. log(X1):log(X8)
## 17. log(X1):log(X9)
## 18. log(X2):log(X3)
## 19. log(X2):log(X4)
## 20. log(X2):log(X5)
## 21. log(X2):log(X6)
## 22. log(X2):log(X7)
## 23. log(X2):log(X8)
## 24. log(X2):log(X9)
## 25. log(X3):log(X4)
## 26. log(X3):log(X5)
## 27. log(X3):log(X6)
## 28. log(X3):log(X7)
## 29. log(X3):log(X8)
## 30. log(X3):log(X9)
## 31. log(X4):log(X5)
## 32. log(X4):log(X6)
## 33. log(X4):log(X7)
## 34. log(X4):log(X8)
## 35. log(X4):log(X9)
## 36. log(X5):log(X6)
## 37. log(X5):log(X7)
## 38. log(X5):log(X8)
## 39. log(X5):log(X9)
## 40. log(X6):log(X7)
## 41. log(X6):log(X8)
## 42. log(X6):log(X9)
## 43. log(X7):log(X8)
## 44. log(X7):log(X9)
## 45. log(X8):log(X9)
##
## We are selecting variables based on p value...
##
## Variables Entered/Removed:
##
## - log(X1):log(X2) added
## - log(X8) added
## - log(X9) added
## - log(X6) added
## - log(X7) added
## - log(X6) added
## - log(X2) added
## - log(X9) added
## - log(X5) added
## - log(X8) added
## - log(X3) added
## - log(X1):log(X2) added
## - log(X1) added
## - log(X7) added
## - log(X1):log(X9) added
## - log(X2) added
## - log(X4) added
## - log(X1):log(X9) added
## - log(X4):log(X8) added
## - log(X4) added
## - log(X3):log(X9) added
## - log(X2):log(X8) added
## - log(X6):log(X7) added
## - log(X2):log(X9) added
##
## No more variables to be added/removed.
##
##
## Final Model Output
## ------------------
##
## Model Summary
## -------------------------------------------------------------
## R 0.996 RMSE 0.171
## R-Squared 0.991 Coef. Var 2.685
## Adj. R-Squared 0.988 MSE 0.029
## Pred R-Squared 0.981 MAE 0.109
## -------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## --------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## --------------------------------------------------------------------
## Regression 70.779 8 8.847 302.669 0.0000
## Residual 0.614 21 0.029
## Total 71.393 29
## --------------------------------------------------------------------
##
## Parameter Estimates
## --------------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## --------------------------------------------------------------------------------------------
## (Intercept) -3.681 2.633 -1.398 0.177 -9.156 1.794
## log(X5) 2.133 0.620 0.141 3.438 0.002 0.843 3.423
## log(X3) 0.690 0.095 0.228 7.286 0.000 0.493 0.887
## log(X1) 0.603 0.048 0.803 12.443 0.000 0.502 0.704
## log(X4):log(X8) 0.358 0.065 0.371 5.490 0.000 0.222 0.494
## log(X3):log(X9) -0.379 0.106 -0.297 -3.565 0.002 -0.601 -0.158
## log(X8):log(X2) -0.219 0.040 -0.280 -5.531 0.000 -0.301 -0.136
## log(X6):log(X7) 0.185 0.053 0.077 3.460 0.002 0.074 0.296
## log(X9):log(X2) 0.133 0.052 0.203 2.554 0.019 0.025 0.241
## --------------------------------------------------------------------------------------------k##
## Stepwise Selection Summary
## ---------------------------------------------------------------------------------------------
## Added/ Adj.
## Step Variable Removed R-Square R-Square C(p) AIC RMSE
## ---------------------------------------------------------------------------------------------
## 1 log(X1):log(X2) addition 0.915 0.912 160.4600 43.2742 0.4662
## 2 log(X8) addition 0.920 0.914 151.7000 43.4911 0.4608
## 3 log(X9) addition 0.970 0.966 44.6280 16.4013 0.2892
## 4 log(X6) addition 0.972 0.967 41.5170 16.0073 0.2834
## 5 log(X7) addition 0.972 0.966 43.4490 17.9741 0.2891
## 6 log(X6) removal 0.970 0.965 46.4730 18.3319 0.2946
## 7 log(X2) addition 0.975 0.970 36.8870 14.5859 0.2732
## 8 log(X9) removal 0.926 0.915 141.2310 44.9129 0.4588
## 9 log(X5) addition 0.929 0.914 137.0590 45.7420 0.4592
## 10 log(X8) removal 0.924 0.912 145.3660 45.6727 0.4646
## 11 log(X3) addition 0.927 0.912 140.6940 46.4372 0.4645
## 12 log(X1):log(X2) removal 0.892 0.875 215.3970 56.2660 0.5543
## 13 log(X1) addition 0.933 0.919 128.7640 44.0925 0.4467
## 14 log(X7) removal 0.933 0.922 126.8600 42.1123 0.4378
## 15 log(X1):log(X9) addition 0.950 0.940 91.2400 35.2341 0.3854
## 16 log(X2) removal 0.949 0.941 90.5010 33.5785 0.3798
## 17 log(X4) addition 0.950 0.939 92.4090 35.5536 0.3875
## 18 log(X1):log(X9) removal 0.932 0.921 128.0910 42.3625 0.4397
## 19 log(X4):log(X8) addition 0.941 0.929 111.3120 40.2947 0.4193
## 20 log(X4) removal 0.940 0.931 110.3630 38.5375 0.4125
## 21 log(X3):log(X9) addition 0.975 0.970 35.7360 13.9503 0.2703
## 22 log(X2):log(X8) addition 0.984 0.980 18.4730 2.6333 0.2212
## 23 log(X6):log(X7) addition 0.989 0.985 10.6510 -5.4275 0.1912
## 24 log(X2):log(X9) addition 0.991 0.988 6.8110 -11.5396 0.1710
## ---------------------------------------------------------------------------------------------# plot(k)
# Stepwise AIC Regression for all log model #
k <- ols_step_both_aic(model_wf_all_log_inter)## Stepwise Selection Method
## -------------------------
##
## Candidate Terms:
##
## 1 . log(X1)
## 2 . log(X2)
## 3 . log(X3)
## 4 . log(X4)
## 5 . log(X5)
## 6 . log(X6)
## 7 . log(X7)
## 8 . log(X8)
## 9 . log(X9)
## 10 . log(X1):log(X2)
## 11 . log(X1):log(X3)
## 12 . log(X1):log(X4)
## 13 . log(X1):log(X5)
## 14 . log(X1):log(X6)
## 15 . log(X1):log(X7)
## 16 . log(X1):log(X8)
## 17 . log(X1):log(X9)
## 18 . log(X2):log(X3)
## 19 . log(X2):log(X4)
## 20 . log(X2):log(X5)
## 21 . log(X2):log(X6)
## 22 . log(X2):log(X7)
## 23 . log(X2):log(X8)
## 24 . log(X2):log(X9)
## 25 . log(X3):log(X4)
## 26 . log(X3):log(X5)
## 27 . log(X3):log(X6)
## 28 . log(X3):log(X7)
## 29 . log(X3):log(X8)
## 30 . log(X3):log(X9)
## 31 . log(X4):log(X5)
## 32 . log(X4):log(X6)
## 33 . log(X4):log(X7)
## 34 . log(X4):log(X8)
## 35 . log(X4):log(X9)
## 36 . log(X5):log(X6)
## 37 . log(X5):log(X7)
## 38 . log(X5):log(X8)
## 39 . log(X5):log(X9)
## 40 . log(X6):log(X7)
## 41 . log(X6):log(X8)
## 42 . log(X6):log(X9)
## 43 . log(X7):log(X8)
## 44 . log(X7):log(X9)
## 45 . log(X8):log(X9)
##
##
## Variables Entered/Removed:
##
## - log(X1):log(X2) added
## - log(X1):log(X9) added
## - log(X1):log(X4) added
## - log(X5):log(X8) added
## - log(X3):log(X9) added
## - log(X1):log(X5) added
## - log(X4):log(X8) added
## - log(X1):log(X9) removed
## - log(X6) added
## - log(X9) added
## - log(X3):log(X8) added
## - log(X3):log(X9) removed
## - log(X4):log(X9) added
## - log(X2):log(X4) added
## - log(X7):log(X9) added
## - log(X7):log(X8) added
##
## No more variables to be added or removed.k##
##
## Stepwise Summary
## ---------------------------------------------------------------------------------
## Variable Method AIC RSS Sum Sq R-Sq Adj. R-Sq
## ---------------------------------------------------------------------------------
## log(X1):log(X2) addition 43.274 6.085 65.308 0.91477 0.91173
## log(X1):log(X9) addition 40.858 5.252 66.141 0.92644 0.92099
## log(X1):log(X4) addition 36.333 4.225 67.167 0.94082 0.93399
## log(X5):log(X8) addition 33.745 3.626 67.766 0.94921 0.94108
## log(X3):log(X9) addition 9.187 1.496 69.896 0.97904 0.97468
## log(X1):log(X5) addition -1.660 0.975 70.418 0.98634 0.98278
## log(X4):log(X8) addition -3.960 0.845 70.548 0.98817 0.98440
## log(X1):log(X9) removal -4.665 0.882 70.510 0.98765 0.98442
## log(X6) addition -7.138 0.760 70.633 0.98936 0.98597
## log(X9) addition -17.032 0.511 70.881 0.99284 0.99011
## log(X3):log(X8) addition -17.687 0.468 70.925 0.99345 0.99050
## log(X3):log(X9) removal -19.540 0.470 70.922 0.99341 0.99091
## log(X4):log(X9) addition -21.843 0.407 70.985 0.99429 0.99173
## log(X2):log(X4) addition -22.522 0.373 71.020 0.99478 0.99204
## log(X7):log(X9) addition -22.985 0.343 71.049 0.99519 0.99226
## log(X7):log(X8) addition -28.798 0.264 71.128 0.99630 0.99368
## ---------------------------------------------------------------------------------# plot(k)
# Stepwise Regression based on p values for all log model #
k <- ols_step_both_p(model_wf_mix2 )## Stepwise Selection Method
## ---------------------------
##
## Candidate Terms:
##
## 1. X1
## 2. X3
## 3. X4
## 4. log(X2)
## 5. log(X5)
## 6. log(X6)
## 7. log(X7)
## 8. log(X8)
## 9. log(X9)
## 10. X1:X3
## 11. X1:X4
## 12. X3:X4
##
## We are selecting variables based on p value...
##
## Variables Entered/Removed:
##
## - X4 added
## - X1:X4 added
## - log(X5) added
## - log(X8) added
## - log(X9) added
## - log(X6) added
## - log(X2) added
## - log(X6) added
## - log(X7) added
## - log(X2) added
## - X3:X4 added
## - X3 added
## - X1:X4 added
## - X1 added
## - X3:X4 added
## - X1:X3 added
## - X3 added
##
##
## Final Model Output
## ------------------
##
## Model Summary
## -------------------------------------------------------------
## R 0.986 RMSE 0.304
## R-Squared 0.971 Coef. Var 4.779
## Adj. R-Squared 0.962 MSE 0.093
## Pred R-Squared 0.948 MAE 0.213
## -------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## --------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## --------------------------------------------------------------------
## Regression 69.356 7 9.908 107.039 0.0000
## Residual 2.036 22 0.093
## Total 71.393 29
## --------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) 20.773 4.570 4.546 0.000 11.295 30.250
## X4 0.480 0.045 2.197 10.600 0.000 0.386 0.574
## log(X5) -4.256 1.077 -0.282 -3.952 0.001 -6.489 -2.023
## log(X8) 1.931 0.238 0.607 8.109 0.000 1.437 2.424
## log(X9) -1.697 0.210 -0.621 -8.086 0.000 -2.132 -1.262
## log(X7) -0.851 0.288 -0.255 -2.953 0.007 -1.449 -0.253
## X1 0.581 0.208 1.031 2.791 0.011 0.149 1.013
## X1:X3 -0.211 0.045 -2.332 -4.648 0.000 -0.305 -0.117
## ----------------------------------------------------------------------------------------k##
## Stepwise Selection Summary
## -------------------------------------------------------------------------------------
## Added/ Adj.
## Step Variable Removed R-Square R-Square C(p) AIC RMSE
## -------------------------------------------------------------------------------------
## 1 X4 addition 0.803 0.796 261.0600 68.4060 0.7087
## 2 X1:X4 addition 0.876 0.867 156.2490 56.4454 0.5719
## 3 log(X5) addition 0.882 0.868 150.2850 57.0898 0.5697
## 4 log(X8) addition 0.889 0.871 141.5510 57.1598 0.5626
## 5 log(X9) addition 0.952 0.942 51.9580 34.0524 0.3779
## 6 log(X6) addition 0.964 0.955 36.1250 27.2246 0.3332
## 7 log(X2) addition 0.966 0.955 35.9080 27.9207 0.3334
## 8 log(X6) removal 0.952 0.940 53.2740 35.7576 0.3841
## 9 log(X7) addition 0.952 0.937 55.2680 37.7548 0.3927
## 10 log(X2) removal 0.952 0.939 53.9410 36.0450 0.3860
## 11 X3:X4 addition 0.971 0.962 27.7710 22.5817 0.3050
## 12 X3 addition 0.987 0.982 7.2280 1.3060 0.2118
## 13 X1:X4 removal 0.980 0.973 15.7040 12.3534 0.2572
## 14 X1 addition 0.985 0.979 10.1640 5.5697 0.2274
## 15 X3:X4 removal 0.943 0.925 68.3730 42.9531 0.4283
## 16 X1:X3 addition 0.984 0.978 10.9560 6.6232 0.2314
## 17 X3 removal 0.971 0.962 27.5690 22.4362 0.3042
## -------------------------------------------------------------------------------------# plot(k)
k <- ols_step_both_aic(model_wf_mix2)## Stepwise Selection Method
## -------------------------
##
## Candidate Terms:
##
## 1 . X1
## 2 . X3
## 3 . X4
## 4 . log(X2)
## 5 . log(X5)
## 6 . log(X6)
## 7 . log(X7)
## 8 . log(X8)
## 9 . log(X9)
## 10 . X1:X3
## 11 . X1:X4
## 12 . X3:X4
##
##
## Variables Entered/Removed:
##
## - X4 added
## - X1:X4 added
## - X3 added
## - X3:X4 added
## - log(X5) added
## - X1:X4 removed
## - log(X8) added
## - log(X9) added
## - X1:X3 added
## - log(X5) removed
## - log(X6) added
## - X1 added
##
## No more variables to be added or removed.k##
##
## Stepwise Summary
## --------------------------------------------------------------------------
## Variable Method AIC RSS Sum Sq R-Sq Adj. R-Sq
## --------------------------------------------------------------------------
## X4 addition 68.406 14.063 57.330 0.80302 0.79599
## X1:X4 addition 56.445 8.830 62.562 0.87632 0.86715
## X3 addition 56.315 8.225 63.168 0.88479 0.87150
## X3:X4 addition 46.745 5.593 65.800 0.92166 0.90913
## log(X5) addition 44.136 4.796 66.596 0.93282 0.91882
## X1:X4 removal 42.233 4.812 66.581 0.93260 0.92182
## log(X8) addition 42.149 4.489 66.904 0.93712 0.92402
## log(X9) addition 10.356 1.455 69.937 0.97962 0.97430
## X1:X3 addition 4.026 1.102 70.290 0.98456 0.97964
## log(X5) removal 3.365 1.153 70.240 0.98385 0.97964
## log(X6) addition 1.186 1.003 70.390 0.98595 0.98148
## X1 addition 0.032 0.903 70.490 0.98735 0.98254
## --------------------------------------------------------------------------# plot(k)
# Stepwise Regression based on p values for X4 eliminated model#
# k <- ols_step_both_p(model_wf_rm4_log)
# k
# plot(k)
# Stepwise AIC Regression for X4 eliminated model#
# k<- ols_step_both_aic(model_wf_rm4_log)
# k
# plot(k)
# Stepwise Regression based on p values for X1 eliminated model#
# k <- ols_step_both_p(model_wf_rm1_log)
# k
# plot(k)
# Stepwise AIC Regression for X1 eliminated model#
# k<- ols_step_both_aic(model_wf_rm1_log)
# k
# plot(k)# All Possible Regression for full log model #
k <- ols_step_all_possible(model_wf_full_log)
# plot(k)
head(arrange(k, desc(adjr)))
# All Possible Regression for all log model #
k <- ols_step_all_possible(model_wf_all_log)
# plot(k)
head(arrange(k, desc(adjr)))
# All Possible Regression for 3g log model #
#!!!!!!!!!!!! k <- ols_step_all_possible(model_wf_3g_log_inter)
# plot(k)
# head(arrange(k, desc(adjr)))
# All Possible Regression for mixed log model #
# k <- ols_step_all_possible(model_wf_mix2 )
# plot(k)
# head(arrange(k, desc(adjr)))
# All Possible Regression for X4 eliminated model #
# k <- ols_step_all_possible(model_wf_rm4_log)
# k
# plot(k)
# All Possible Regression for X1 eliminated model #
# k <- ols_step_all_possible(model_wf_rm1_log)
# k
# plot(k)#Lack of Fit F Test
ols_pure_error_anova(lm(y~X1, data = table_wf))
ols_pure_error_anova(lm(y~X4, data = table_wf))
alias(lm(y ~ as.factor(X3) + as.factor(X4) + as.factor(X5) + as.factor(X6) + as.factor(X7), data=table_wf))
alias(lm(y ~ as.factor(X1) + as.factor(X8) , data=table_wf))
alias(lm(y ~ as.factor(X4) + as.factor(X9) , data=table_wf))
alias(lm(y ~ as.factor(X3) + as.factor(X6) + as.factor(X7) + as.factor(X8) + as.factor(X9) , data=table_wf))Final models
- Check model aic_all_log
ols_regress(model_wf_aic_all_log )## Model Summary
## -------------------------------------------------------------
## R 0.993 RMSE 0.208
## R-Squared 0.986 Coef. Var 3.273
## Adj. R-Squared 0.982 MSE 0.043
## Pred R-Squared 0.975 MAE 0.136
## -------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## --------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## --------------------------------------------------------------------
## Regression 70.394 6 11.732 270.106 0.0000
## Residual 0.999 23 0.043
## Total 71.393 29
## --------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) 0.571 3.360 0.170 0.866 -6.379 7.522
## log(X1) 0.726 0.036 0.967 20.107 0.000 0.651 0.800
## log(X3) 0.419 0.096 0.139 4.359 0.000 0.220 0.617
## log(X5) 1.259 0.796 0.083 1.582 0.127 -0.387 2.905
## log(X6) -0.267 0.090 -0.087 -2.960 0.007 -0.454 -0.080
## log(X8) 1.623 0.175 0.510 9.267 0.000 1.260 1.985
## log(X9) -1.375 0.154 -0.503 -8.919 0.000 -1.694 -1.056
## ----------------------------------------------------------------------------------------summary(model_wf_aic_all_log)##
## Call:
## lm(formula = log(y) ~ log(X1) + log(X3) + log(X5) + log(X6) +
## log(X8) + log(X9), data = table_wf)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.67722 -0.08003 0.01102 0.13879 0.25715
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.57120 3.36000 0.170 0.86650
## log(X1) 0.72550 0.03608 20.107 4.31e-16 ***
## log(X3) 0.41866 0.09605 4.359 0.00023 ***
## log(X5) 1.25873 0.79566 1.582 0.12731
## log(X6) -0.26702 0.09022 -2.960 0.00702 **
## log(X8) 1.62253 0.17508 9.267 3.15e-09 ***
## log(X9) -1.37489 0.15416 -8.919 6.33e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2084 on 23 degrees of freedom
## Multiple R-squared: 0.986, Adjusted R-squared: 0.9824
## F-statistic: 270.1 on 6 and 23 DF, p-value: < 2.2e-16Anova(model_wf_aic_all_log)| Sum Sq | Df | F value | Pr(>F) |
| 17.6 | 1 | 404 | 4.31e-16 |
| 0.825 | 1 | 19 | 0.00023 |
| 0.109 | 1 | 2.5 | 0.127 |
| 0.38 | 1 | 8.76 | 0.00702 |
| 3.73 | 1 | 85.9 | 3.15e-09 |
| 3.45 | 1 | 79.5 | 6.33e-09 |
| 0.999 | 23 |
# Collinearity Diagnostics #
ols_vif_tol(model_wf_aic_all_log)| Variables | Tolerance | VIF |
| log(X1) | 0.263 | 3.8 |
| log(X3) | 0.603 | 1.66 |
| log(X5) | 0.22 | 4.55 |
| log(X6) | 0.71 | 1.41 |
| log(X8) | 0.201 | 4.99 |
| log(X9) | 0.191 | 5.22 |
#Model Fit Assessment
ols_plot_diagnostics(model_wf_aic_all_log)
# Part & Partial Correlations
ols_test_correlation(model_wf_aic_all_log) # Correlation between observed residuals and expected residuals under normality.## [1] 0.9278999# Residual Normality Test
ols_test_normality(model_wf_aic_all_log) # Test for detecting violation of normality assumption. #If p-value is bigger, then no problem of non-normality ### -----------------------------------------------
## Test Statistic pvalue
## -----------------------------------------------
## Shapiro-Wilk 0.8746 0.0021
## Kolmogorov-Smirnov 0.0964 0.9180
## Cramer-von Mises 7.0221 0.0000
## Anderson-Darling 0.7277 0.0516
## -----------------------------------------------# Variable Contributions
ols_plot_added_variable(model_wf_aic_all_log)
# Residual Plus Component Plot
ols_plot_comp_plus_resid(model_wf_aic_all_log)
- Check model aic_log_inter
summary(model_wf_aic_log_inter)##
## Call:
## lm(formula = log(y) ~ X1 + X2 + X3 + X4 + X5 + X7 + X8 + X9 +
## X1:X3 + X1:X9 + X2:X9 + X3:X8 + X4:X8 + X7:X9, data = table_wf)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.22469 -0.06872 0.00151 0.04884 0.42819
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.98138 1.51954 -0.646 0.528142
## X1 2.71181 0.32987 8.221 6.15e-07 ***
## X2 -24.45201 5.63070 -4.343 0.000580 ***
## X3 0.49272 0.06581 7.487 1.93e-06 ***
## X4 0.16780 0.05415 3.099 0.007335 **
## X5 0.03903 0.01606 2.430 0.028137 *
## X7 2.51458 1.25764 1.999 0.064011 .
## X8 0.58605 0.08651 6.775 6.26e-06 ***
## X9 -0.24828 0.10461 -2.373 0.031416 *
## X1:X3 -0.41634 0.04505 -9.242 1.40e-07 ***
## X1:X9 -0.11820 0.03621 -3.264 0.005231 **
## X2:X9 4.64270 1.44938 3.203 0.005925 **
## X3:X8 -0.05085 0.01367 -3.721 0.002050 **
## X4:X8 0.07075 0.01481 4.777 0.000244 ***
## X7:X9 -0.99460 0.34359 -2.895 0.011113 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1711 on 15 degrees of freedom
## Multiple R-squared: 0.9939, Adjusted R-squared: 0.9881
## F-statistic: 173.2 on 14 and 15 DF, p-value: 5.77e-14Anova(model_wf_aic_log_inter)| Sum Sq | Df | F value | Pr(>F) |
| 0.586 | 1 | 20 | 0.000446 |
| 0.277 | 1 | 9.48 | 0.00764 |
| 0.00823 | 1 | 0.281 | 0.604 |
| 2.99 | 1 | 102 | 4.36e-08 |
| 0.173 | 1 | 5.9 | 0.0281 |
| 0.00974 | 1 | 0.333 | 0.573 |
| 1.88 | 1 | 64.3 | 8.37e-07 |
| 2.59 | 1 | 88.4 | 1.12e-07 |
| 2.5 | 1 | 85.4 | 1.4e-07 |
| 0.312 | 1 | 10.7 | 0.00523 |
| 0.3 | 1 | 10.3 | 0.00592 |
| 0.405 | 1 | 13.8 | 0.00205 |
| 0.668 | 1 | 22.8 | 0.000244 |
| 0.245 | 1 | 8.38 | 0.0111 |
| 0.439 | 15 |
# Collinearity Diagnostics #
ols_vif_tol(model_wf_aic_log_inter)| Variables | Tolerance | VIF |
| X1 | 0.0012 | 835 |
| X2 | 0.032 | 31.3 |
| X3 | 0.0141 | 70.8 |
| X4 | 0.00668 | 150 |
| X5 | 0.0824 | 12.1 |
| X7 | 0.0262 | 38.1 |
| X8 | 0.0842 | 11.9 |
| X9 | 0.0371 | 27 |
| X1:X3 | 0.00165 | 605 |
| X1:X9 | 0.00557 | 179 |
| X2:X9 | 0.0287 | 34.9 |
| X3:X8 | 0.024 | 41.7 |
| X4:X8 | 0.00579 | 173 |
| X7:X9 | 0.0111 | 90.3 |
#Model Fit Assessment
ols_plot_diagnostics(model_wf_aic_log_inter)
# Part & Partial Correlations
ols_test_correlation(model_wf_aic_log_inter) # Correlation between observed residuals and expected residuals under normality.## [1] 0.9380444# Residual Normality Test
ols_test_normality(model_wf_aic_log_inter) # Test for detecting violation of normality assumption. #If p-value is bigger, then no problem of non-normality ### -----------------------------------------------
## Test Statistic pvalue
## -----------------------------------------------
## Shapiro-Wilk 0.898 0.0075
## Kolmogorov-Smirnov 0.1576 0.4037
## Cramer-von Mises 8.0824 0.0000
## Anderson-Darling 0.8459 0.0259
## -----------------------------------------------# Variable Contributions
ols_plot_added_variable(model_wf_aic_log_inter)
# Residual Plus Component Plot
ols_plot_comp_plus_resid(model_wf_aic_log_inter)
- Check model mixed
summary(model_wf_aic_mix2 )##
## Call:
## lm(formula = log(y) ~ X1 + X3 + X4 + log(X7) + log(X8) + log(X9) +
## X1:X3 + X1:X4, data = table_wf)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.56967 -0.05648 -0.01439 0.08018 0.33666
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.04446 0.34314 5.958 6.50e-06 ***
## X1 1.45925 0.23161 6.301 3.01e-06 ***
## X3 0.27464 0.03851 7.131 4.94e-07 ***
## X4 0.42303 0.04203 10.066 1.73e-09 ***
## log(X7) -0.42799 0.20817 -2.056 0.0524 .
## log(X8) 1.62847 0.16269 10.010 1.90e-09 ***
## log(X9) -1.40518 0.14700 -9.559 4.24e-09 ***
## X1:X3 -0.39830 0.05278 -7.546 2.07e-07 ***
## X1:X4 0.02624 0.01181 2.222 0.0374 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2116 on 21 degrees of freedom
## Multiple R-squared: 0.9868, Adjusted R-squared: 0.9818
## F-statistic: 196.7 on 8 and 21 DF, p-value: < 2.2e-16Anova(model_wf_aic_mix2 )| Sum Sq | Df | F value | Pr(>F) |
| 0.372 | 1 | 8.32 | 0.00888 |
| 0.167 | 1 | 3.72 | 0.0674 |
| 5.54 | 1 | 124 | 2.93e-10 |
| 0.189 | 1 | 4.23 | 0.0524 |
| 4.49 | 1 | 100 | 1.9e-09 |
| 4.09 | 1 | 91.4 | 4.24e-09 |
| 2.55 | 1 | 56.9 | 2.07e-07 |
| 0.221 | 1 | 4.94 | 0.0374 |
| 0.94 | 21 |
# Collinearity Diagnostics #
ols_vif_tol(model_wf_aic_mix2)| Variables | Tolerance | VIF |
| X1 | 0.00371 | 269 |
| X3 | 0.0631 | 15.9 |
| X4 | 0.017 | 59 |
| log(X7) | 0.161 | 6.2 |
| log(X8) | 0.239 | 4.18 |
| log(X9) | 0.217 | 4.61 |
| X1:X3 | 0.00184 | 543 |
| X1:X4 | 0.00401 | 249 |
#Model Fit Assessment
ols_plot_diagnostics(model_wf_aic_mix2)
# Part & Partial Correlations
ols_test_correlation(model_wf_aic_mix2) # Correlation between observed residuals and expected residuals under normality.## [1] 0.9628432# Residual Normality Test
ols_test_normality(model_wf_aic_mix2) # Test for detecting violation of normality assumption. #If p-value is bigger, then no problem of non-normality ### -----------------------------------------------
## Test Statistic pvalue
## -----------------------------------------------
## Shapiro-Wilk 0.9393 0.0868
## Kolmogorov-Smirnov 0.1411 0.5422
## Cramer-von Mises 7.0915 0.0000
## Anderson-Darling 0.5183 0.1733
## -----------------------------------------------# Variable Contributions
ols_plot_added_variable(model_wf_aic_mix2)
# Residual Plus Component Plot
ols_plot_comp_plus_resid(model_wf_aic_mix2)
- Check model all log interaction
summary(model_wf_aic_all_log_inter)##
## Call:
## lm(formula = log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) +
## log(X5) + log(X6) + log(X7) + log(X8) + log(X9) + log(X1):log(X3) +
## log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X8) + log(X2):log(X9) +
## log(X3):log(X9) + log(X4):log(X8) + log(X4):log(X9) + log(X5):log(X8) +
## log(X5):log(X9) + log(X6):log(X8) + log(X6):log(X9) + log(X7):log(X9),
## data = table_wf)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.114032 -0.038669 -0.003953 0.026220 0.160039
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -16.0272 13.9469 -1.149 0.28823
## log(X1) 1.1577 0.6966 1.662 0.14045
## log(X2) -0.7328 0.3383 -2.166 0.06698 .
## log(X3) 1.6656 1.7630 0.945 0.37625
## log(X4) -2.3936 2.9633 -0.808 0.44581
## log(X5) 5.0279 3.4728 1.448 0.19094
## log(X6) -0.1583 0.3004 -0.527 0.61463
## log(X7) -0.4075 0.5183 -0.786 0.45748
## log(X8) 44.9189 24.1746 1.858 0.10550
## log(X9) -37.0656 19.1702 -1.934 0.09443 .
## log(X1):log(X3) 0.8234 0.7067 1.165 0.28218
## log(X1):log(X8) 1.2421 0.6407 1.939 0.09372 .
## log(X1):log(X9) -0.9995 0.6739 -1.483 0.18156
## log(X2):log(X8) 1.0890 0.3972 2.742 0.02885 *
## log(X2):log(X9) -0.7095 0.3415 -2.078 0.07633 .
## log(X3):log(X9) 0.4112 0.3562 1.154 0.28626
## log(X4):log(X8) -3.0288 0.8494 -3.566 0.00915 **
## log(X4):log(X9) 2.1744 0.9286 2.342 0.05172 .
## log(X5):log(X8) -7.9687 5.5620 -1.433 0.19504
## log(X5):log(X9) 6.7951 4.4498 1.527 0.17059
## log(X6):log(X8) 0.4027 0.2891 1.393 0.20622
## log(X6):log(X9) -0.5057 0.2651 -1.908 0.09807 .
## log(X7):log(X9) 0.4637 0.3759 1.234 0.25719
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1252 on 7 degrees of freedom
## Multiple R-squared: 0.9985, Adjusted R-squared: 0.9936
## F-statistic: 206.6 on 22 and 7 DF, p-value: 7.713e-08Anova(model_wf_aic_all_log_inter)| Sum Sq | Df | F value | Pr(>F) |
| 0.281 | 1 | 17.9 | 0.00386 |
| 0.00972 | 1 | 0.619 | 0.457 |
| 0.0102 | 1 | 0.648 | 0.447 |
| 0.000172 | 1 | 0.0109 | 0.92 |
| 0.0311 | 1 | 1.98 | 0.202 |
| 0.068 | 1 | 4.33 | 0.0759 |
| 0.00234 | 1 | 0.149 | 0.711 |
| 1.7 | 1 | 108 | 1.65e-05 |
| 1.73 | 1 | 111 | 1.53e-05 |
| 0.0213 | 1 | 1.36 | 0.282 |
| 0.059 | 1 | 3.76 | 0.0937 |
| 0.0345 | 1 | 2.2 | 0.182 |
| 0.118 | 1 | 7.52 | 0.0288 |
| 0.0677 | 1 | 4.32 | 0.0763 |
| 0.0209 | 1 | 1.33 | 0.286 |
| 0.199 | 1 | 12.7 | 0.00915 |
| 0.086 | 1 | 5.48 | 0.0517 |
| 0.0322 | 1 | 2.05 | 0.195 |
| 0.0366 | 1 | 2.33 | 0.171 |
| 0.0304 | 1 | 1.94 | 0.206 |
| 0.0571 | 1 | 3.64 | 0.0981 |
| 0.0239 | 1 | 1.52 | 0.257 |
| 0.11 | 7 |
# Collinearity Diagnostics #
ols_vif_tol(model_wf_aic_all_log_inter)| Variables | Tolerance | VIF |
| log(X1) | 0.000255 | 3.92e+03 |
| log(X2) | 0.00469 | 213 |
| log(X3) | 0.000646 | 1.55e+03 |
| log(X4) | 4.36e-05 | 2.29e+04 |
| log(X5) | 0.00416 | 240 |
| log(X6) | 0.0231 | 43.2 |
| log(X7) | 0.00911 | 110 |
| log(X8) | 3.8e-06 | 2.63e+05 |
| log(X9) | 4.47e-06 | 2.24e+05 |
| log(X1):log(X3) | 0.000108 | 9.3e+03 |
| log(X1):log(X8) | 0.000275 | 3.64e+03 |
| log(X1):log(X9) | 0.000202 | 4.94e+03 |
| log(X2):log(X8) | 0.000851 | 1.18e+03 |
| log(X2):log(X9) | 0.000809 | 1.24e+03 |
| log(X3):log(X9) | 0.00282 | 355 |
| log(X4):log(X8) | 0.000283 | 3.53e+03 |
| log(X4):log(X9) | 0.00019 | 5.25e+03 |
| log(X5):log(X8) | 4.17e-06 | 2.4e+05 |
| log(X5):log(X9) | 4.82e-06 | 2.07e+05 |
| log(X6):log(X8) | 0.0203 | 49.4 |
| log(X6):log(X9) | 0.0191 | 52.3 |
| log(X7):log(X9) | 0.00632 | 158 |
#Model Fit Assessment
ols_plot_diagnostics(model_wf_aic_all_log_inter)
# Part & Partial Correlations
ols_test_correlation(model_wf_aic_all_log_inter) # Correlation between observed residuals and expected residuals under normality.## [1] 0.9830352# Residual Normality Test
ols_test_normality(model_wf_aic_all_log_inter) # Test for detecting violation of normality assumption. #If p-value is bigger, then no problem of non-normality ### -----------------------------------------------
## Test Statistic pvalue
## -----------------------------------------------
## Shapiro-Wilk 0.9695 0.5254
## Kolmogorov-Smirnov 0.1176 0.7579
## Cramer-von Mises 8.8358 0.0000
## Anderson-Darling 0.3654 0.4134
## -----------------------------------------------# Variable Contributions
ols_plot_added_variable(model_wf_aic_all_log_inter)
# Residual Plus Component Plot
ols_plot_comp_plus_resid(model_wf_aic_all_log_inter)
